I'd recommend first that you and your friend spend more time with Tits :), "Reductive groups over local fields", from the Corvallis volume (free online, last time I checked). Undoubtably there are other references, like papers of Prasad-Raghunathan mentioned by Greg Kuperberg, and any paper that treats Bruhat-Tits theory for unitary groups. I'll try to provide a background/basic treatment here.
You're certainly used to Bruhat-Tits theory for $SL_2$, or at least for $PGL_2$, over $Q_p$, where one encounters the $(p+1)$-regular tree. As you know, $PGL_2(Q_p)$ acts on this tree, and the stabilizer of a point is a maximal compact subgroup that is conjugate to $PGL_2(Z_p)$, and the stabilizer of an edge is an Iwahori subgroup. This I assume is a familiar picture.
To understand the "special point" subtleties, you should first think about a group like $G=SU_3$ -- the $Q_p$-points of a quasisplit form of $SL_3$, associated to an unramified quadratic field extension $K/Q_p$ with $p$ odd. Let $S$ be a maximal $Q_p$-split torus in $G$. Then $S$ has rank one, though the group $G$ has absolute rank two. It follows that the Bruhat-Tits building for $SU_3$ is again a tree, though not as simple as the $SL_2$ case. In fact, in this unramified situation, the building can be seen as the fixed points of the building of $G$ over $K$ (which is the building of $SL_3$), under the Galois involution.
Now, one must think about the relative roots of $G$ with respect to $S$ -- i.e., decompose the Lie algebra of $G$ with respect to the adjoint action of $S$. There are four eigenspaces with nontrivial eigenvalue -- these are the root spaces for the relative roots which I'll call $\pm \alpha$ and $\pm 2 \alpha$. The root spaces ${\mathfrak g}_{\pm \alpha}$ are two-dimensional.
Now, let $A$ be the apartment of the building associated to $S$ -- $A$ is a principal homogeneous space for the one-dimensional real vector space $X_\bullet(S) \otimes_Z R$. After choosing a good base point (a hyperspecial base point, using the fact that $K/Q_p$ is unramified), $A$ may be identified with $X_\bullet(S) \otimes_Z R$ and the affine roots are functions of the form $\pm \alpha + k$ and $\pm 2 \alpha + k$, where $k$ can be any integer.
Let $h$ be a generator of the rank 1 $Z$-module $X_\bullet(S)$, so that $\alpha^\vee = 2 h$, and $A = R \cdot h$. The affine roots are given by:
$$[\pm \alpha + k](r h) = r + k, [\pm 2 \alpha + k](r h) = 2r + k.$$
The vanishing hyperplanes of these affine roots are the points:
$$r h : r \in \frac{1}{2} Z.$$
These are the vertices of the building, contained in the apartment $A$.
Now consider a vertex $nh$, where $n$ is an integer. The affine roots $\pm (\alpha - n)$ and $\pm (2 \alpha - 2n)$ vanish at the vertex $n$. The gradients of these affine roots are the roots $\pm \alpha$ and $\pm 2 \alpha$. These are all of the roots in the original (relative) root system. That's why these vertices are hyperspecial vertices.
On the other hand, consider a vertex $(n + \frac{1}{2}) h$, where $n$ is an integer. The affine roots vanishing at this vertex are $\pm (2 \alpha - 2n - 1)$. The gradients of these affine roots are the roots $\pm 2 \alpha$. These are not all of the original roots, but all original roots are proportional to these roots. You can see how this phenomenon requires the setting of a non-reduced root system to happen. These "half-integral" vertices are special points, since the original root system does not occur in the system of gradients, but it does up to proportionality. At these special (but not hyperspecial) points, the Bruhat-Tits group scheme underlying the parahoric has special fibre with reductive quotient isomorphic to $PGL_2$ (I think... or is it $SL_2$) over the residue field. At the hyperspecial points, the group would be a quasisplit $SU_3$ over the residue field.
If it's not clear from above, a special point in the building occurs where the set of gradients of affine roots vanishing at that point is equal, modulo proportionality, to the set of relative roots. That's the general definition.
Hope this helps - see Tits for more.
(I write this answer in quite a haste, so there will probably be some inaccuracies. Sorry for that and, please, let me know, when you find them.)
First, I think, that if the group is not almost simple, then the question makes few sense. For example, each of the split groups $G_1 = Sp_4(F)$ and $G_2 = {\rm SL}_2(F) \times {\rm SL}_2(F)$ have a parahoric subgroup of the same type, namely ${\rm SL}_2(\mathcal{O}_F) \times {\rm SL}_2(\mathcal{O}_F)$ (where $\mathcal{O}_F$ are the integers of $F$), obviously with the same finite reductive quotient. For $G_1$, the corresponding parahoric is non-special, whereas for $G_2$ it is hyperspecial.
For simplicity, I will even assume that $G$ is absolutely almost simple over $F$, which means that the absolute Dynkin diagram is connected. We have the affine Dynkin diagram $\widetilde \Delta$ of $G(F)$, whose vertices are in 1-1 bijection with the vertices in a fixed base alcove of the Bruhat--Tits building of $G(F)$. Alternatively, the affine Dynkin diagram can be obtained from the usual one by adding one vertex. Conjugacy classes of maximal parahorics $K = K_x$ of $G(F)$ are in 1-1 bijection with vertices $x$ of $\widetilde \Delta$ (use that $G$ is simply connected). The Dynkin diagram of the finite reduction quotient $\mathcal{K}_x = K_x/K_x^+$ is then just $\widetilde \Delta \backslash \{x\}$ (with all edges beginning/ending in $v$ removed). Thus the number of simple factors of $\mathcal{K}_x$ is equal to the number of connected components of $\widetilde \Delta \backslash \{x\}$. For example, if you remove the middle point in (affine) type $E_6$ (but also in some other cases) it is possible for $\widetilde \Delta \backslash \{x\}$ to have $3$ connected components, so that there are three factors.
Now, $x$ is special, iff $\widetilde \Delta \backslash \{x\}$ equals $\Delta$, the (usual, not affine) Dynkin diagram of $G$. To answer the question, observe (by looking at the finite and affine Dynkin diagrams, which can be found for example in wikipedia) that $\widetilde \Delta \backslash \{x\}$ will tend to be of type $\Delta$ when $x$ tends to be an extremal vertex of $\widetilde \Delta$. For example, if $G = Sp_4$, i.e. with $\Delta$ of type $C_2$ and $\widetilde \Delta$ being $\cdot = \cdot = \cdot$, then $\widetilde \Delta$ without the left or the right vertex will be of type $C_2$ (so these are hyperspecial), whereas $\widetilde \Delta$ without the middle vertex is of type $A_1 \times A_1$ (and it is not hyperspecial).
Best Answer
I'm not sure what it means to define parahoric subgroups "purely in terms of $B(G, F)$"; I would say that every definition boils down to taking integral points of integral models in one way or another. By the way, it is not true in general that parahoric subgroups are full facet stabilisers; in general, the group scheme $\mathcal G$ underlying the full stabiliser is disconnected, and one must pass to its identity component before taking integral points in order to get the parahoric. (See nonetheless, say, Tits §3.5.3, or Proposition 4.6.32 of BT2, where it is observed that one does have equality for simply connected groups.)
Nonetheless, the result you want (that parahoric subgroups are pullbacks of parabolic subgroups of parahoric subgroups) is correct; it is §3.5.4 of Tits's Corvallis article "Reductive groups over local fields" (MSN), and Théorème 4.6.33 of BT2 (MSN). Of course, it doesn't really 'reduce' the problem, since one still needs to have the original parahoric subgroup to pull back its parabolic subgroups. You may also find it helpful to read Yu's various expository articles (say, "Bruhat–Tits theory and buildings" (MSN) in the Ottawa proceedings, or his paper "Smooth models associated to concave functions in Bruhat–Tits theory" (MSN; I have linked to the preprint at NUS, which I have not compared to the published version)) for a modern perspective on BT theory; he had a program for a while to make their work more accessible. He used to have some notes available on his Purdue web page, but that no longer exists, and he doesn't seem to have migrated them to CUHK.