Hi T_Co.
I have used the concave mirror from a woman's makeup kit to light a fire a few times. It was only a little bit awkward to find the focal length and start the tinder burning, but it did start a fire very quickly on a sunny day.
It was not front coated and it was smaller than 3 inches.
I doubt if being strictly parabolic instead of spherical matters much on something so small. You need to aim it a bit sideways at the tinder you are holding in your hand because if you don't you end up blocking the sun with your hand.
It was not as compact as a magnifying glass but it was in a compact.

Now you have me wondering about a ladies powder compact with the magnifying mirror as a fire starter kit. An empty compact would have space for tinder, matches, maybe a sparklite and of course a magnifying mirror in the lid. I bet a Fresnel lens would fit in it too.