Why is it that wherever we see spherical mirrors, we mainly find concave and convex mirrors? Why don't we use parabolic mirrors as extensively as the other two types of mirrors? It even has an advantage that any type of incident parallel light ray passes through its focus, unlike concave and convex mirrors where the rays have to be paraxial.
Using parabolic mirrors instead of regular concave and convex mirrors
opticsreflection
Related Solutions
Assuming mirror to be spherical section. C is the center of sphere.
See, Using trigonometry. $$x=d \times \sin(2\theta)$$ $$x=R\times\sin\theta$$
Eliminate $\theta$ and get $d$ : distance from Center of curvature as a function of $x$.
Verify for small theta where $\sin\theta\approx\theta$
If you just want to see that which side ray bents then see. $$d=\dfrac{R\times \sec\theta}2$$ which shows that $d\ge R/2$ . So, ray bends towards the pole as looses it paraxial character.
It is customary to define the focal point of a spherical mirror as that point on the principal axis where rays which are near to and parallel to the principal axis meet.
The reason for including the words near to can be shown as follows.
In the diagram below there is a concave mirror of radius $R$ and centre $C$.
The principal axis is $CP$ and the incident ray, parallel to the principal axis , is shown as $AB$.
Triangles $ACB$ and $CFB$ are similar so $$\dfrac {AB}{AC}= \dfrac {CB}{CF} \Rightarrow \dfrac {AB}{R}= \dfrac {R}{CF}$$
As the incident ray gets closer to the principal axis $AB \to 2R$ and so $CF \to \dfrac R2$.
So it is really all to do with how close the length of $AB$ to a diameter $2R$.
Using such a construction you could quantify the words near to by deciding how different $CF$ can be from $\dfrac R2$.
A parabola $y=\frac 14 x^2$ (blue) and a circle $(y-2)^2+x^2=4$ (red) are shown in the diagram below.
Curvature of both graphs is the same at point $P$ as is the centre of curvature $C_{\rm circle}$.
Just by eye you can see that the two graphs are "very" similar close to $P$.
All rays parallel to the principal axis arrive at $F$ at the same time because $BF= BD$ and so this is the focal pont of the parabola.
There is however a restriction in that this will only happen if the incoming parallel rays are parallel to the principal axis.
The position of the centre of curvature of the parabola $C_{{\rm parabola},B}$ depends on the position of $B$ and only coincides with that of the circle at position $P$.
The necessary construction to find the centre of curvature os a parabola is shown below and more details can be found here from where the gif image was taken.
Best Answer
Besides the obvious reason that a parabolic mirror is much more difficult and expensive to manufacture than a spherical one, a parabolic mirror has the worst coma imaginable (ok, this may be an exaggeration) for off-axis rays, see https://en.wikipedia.org/wiki/Coma_(optics). Your statement that "[...] has an advantage that any type of parallel light ray on it passes through its focus" is true only for rays parallel with the mirror axis, not ones that fall obliquely. Becuase of its symmetry spherical mirror has the same aberration for oblique rays as for rays that are parallel with the axis.