I understand polishing the mirror with a successively finer abrasive can create a curvature, but how is it possible to obtain a uniform curve that forms a parabola while polishing by hand?
How is it ensured that a parabolic shape forms while polishing a telescope mirror?
While a spherical shape is fairly easy to abtain, a parabolic shape definitely isn't!
It's a combination of special polishing strokes, and feedback via control and testing.
Normal polishing strokes (center over center, 1/3 diameter amplitude) tend to make a sphere. When all goes well, the surface of the mirror tends to become spherical, or very close to it. This is not different from grinding, only more precise. When two surfaces are rubbing against each other in a ball-and-socket type of joint, any irregularities will be removed and both surfaces will "want" to become spherical, or close to it.
When making telescope mirrors, the so-called "normal stroke" is what accomplishes that in the best way.
Now here's the essential detail: no matter what other curve you're trying to obtain (parabola, hyperbola, ellipse), for the purpose of making telescope mirrors all these curves are very close to a circle, within a fraction of a micron.
So a typical process has an initial goal of making a sphere (cross-section is a circle). This is not mandatory, but it's easier to first make a sphere, and then apply "corrections" to it. You can definitely make a parabola directly, but the process is more complex.
For telescope mirrors, the parabola that's needed is basically a circle that's deeper in the center, and more shallow towards the edge. So, if you start with a spherical mirror (circular section) and use polishing strokes that tend to deepen the center in a gradual way, you can get pretty close to a parabola. This is known as a "parabolizing stroke".
Keep in mind that as you apply that stroke, the center keeps deepening. At some point the correction will become deeper than a parabola and the mirror will become hyperbolic. That's sometimes what you actually need (if you make a hyperbolic mirror for a Ritchey-Chretien telescope). Or if you stop too early, the mirror is elliptic, which is used in other designs. The parabola is the thin edge between the elliptic and hyperbolic domains.
You start with a mirror where the cross-section is a circle (spherical mirror). You begin to apply the W stroke. The circle immediately becomes a very shallow ellipse, which keeps deepening. At some point, for a very short time, that ellipse becomes a parabola. As you keep doing the W stroke, the parabola then changes quickly and now you have a hyperbola which just becomes more and more deep.
The key here is testing. You apply the parabolizing stroke for a short while, then stop and test the mirror. There are many testing techniques; the Couder mask is simple enough and very popular; or you could use an interferometer if you have (or can build) one. Is it still elliptic? Keep parabolizing. Has it become hyperbolic? Reset to a sphere and try again. Or adjust the stroke if either the edge or the center are being corrected too fast.
Apply parabolizing strokes. Test. Modify technique if necessary. Repeat. That's all there is to it.
Note: I am using jargon in the text above. "Deeper" really means "more strongly curved". "Shallower" really means "less strongly curved". The difference between circle and parabola is that the parabola is more strongly curved in the center, while the curvature at the edge is less strong. The circle has constant curvature everywhere, by definition.
This doesn't necessarily mean the center of the mirror is physically more depressed; while doing any strokes, the whole surface keeps being polished down all the time. But the relative curvatures of center vs. edge may change.
Deeper vs shallower are terms that reflect the physical process, whether the tool applies more or less pressure on a certain area of the mirror (so polishing goes faster or slower there). But the geometric attribute that the process actually changes, that we are interested in, is the radius of curvature. Where the tool applies more pressure, the radius of the curvature decreases rapidly (the surface becomes more strongly curved).
Deeper and shallower are practical terms related to process. More or less strongly curved are theoretical terms related to geometry. When you make mirrors, after a while these terms become linked in your mind.
It helps if you're familiar with the conic sections:
red = circle
orange = ellipse
green = parabola
blue = hyperbola