First make sure all your screws are tight, and that there isn't any shaking because of slack in any areas where things connect to each other.
Another thing you can do is buy vibration dampening pads to put your tripod on.
Finally, you can add counter weights and pendulum weights to the tripod to give it more mass to withstand the wind and touches.
It's a pretty complex topic. Bottom line: if the scope is doing okay, it's probably parabolic.
The one way to tell for sure, and even ascertain figuring errors and whatnot, is to do a proper optical test. If I was in that situation, I would just take the mirror out, put it on the Foucault / Ronchi tester, and take some measurements. A ronchigram would give a qualitative answer immediately: is it a parabola-like curve or not? Are there any gross errors or not? Then a quantitative Foucault test could actually provide some numbers.
You could google the Foucault test and the Ronchi test for telescope mirrors if you're curious.
With the mirror installed in the scope, it's a bit more complex. One thing is certain: at an f/5 focal ratio, if it's not parabolic, performance would be pretty terrible. Even if it is parabolic, this being cheap mass-produced optics, some issues may remain. Even assuming a very expensive, near-perfect parabola, at f/5 an aberration called coma will be pretty obvious (it's a defect of this geometry, still exists even with perfect optics).
One thing you could do is learn star testing. It's not trivial, but it could reveal various issues. Google this topic, it's pretty big. Some links to get you started:
http://legault.perso.sfr.fr/collim.html
http://www.astrosurf.com/altaz/startest_e.htm
One outcome of star testing is that you'll get your scope collimated really really well, which is something you should do anyway. Once that's done, start looking for spherical aberration. A good parabola will show only a residual amount of spherical aberration; a top-shelf perfect parabola will show zero spherical aberration. At f/5, a pure sphere will show horrible amounts of this aberration.
At first, you'll be liable to confuse spherical aberration with turned down edge (assuming your scope exhibits both). Also, the central obstruction (the edge of the secondary mirror) will distort the diffraction figure in ways that could be mistaken for aberration, if you're not very familiar with it. Keep experimenting and learning and you'll figure it out eventually.
Use a good quality, strong eyepiece for the star test. Keep the image in the exact center, to reduce extra issues with the eyepiece. Polaris is a good star for testing, since it doesn't move and it's bright enough. Don't do the test when seeing is bad and the diffraction figure is shaking like a bowl of jello.
Best Answer
An erecting prism will both flip the image right side up and restore left-right orientation. The problem with using one in a reflecting telescope is that because they're several inches long, you'll need to rack your focuser several inches farther in in order to place the eye-piece at the correct location to bring the image into focus. In many cases there won't be enough available infocus to do this unless you modify the telescope to bring the primary mirror closer to the secondary mirror/focuser assemblies (and then you might need a larger secondary to keep wide angle eyepieces fully illuminated at the edge). This isn't a problem for refracting telescopes since it would replace an existing 45/90* diagonal which would have a similar length.