There is a confusion in your question. ALL reflecting telescopes ideally should have parabolic mirrors. The problem is that a parabolic mirror is difficult to make. So they were
made with spherical or approximately parabolic mirrors.
All this has nothing to do with the difference between Newtonian, Gregorian and Cassegrain telescopes. The distinction between these three is in the optical scheme.
Newtonian uses one concave and one flat mirror (or just one concave).
Gregorian uses two concave mirrors, and
Cassegrain uses one concave and one convex mirror.
All these mirrors should be parabolic in an ideal situation. First Gregorian was made by Hooke in 1673.
This was 5 years before John Hadley was born. Hadley made a parabolic mirror in 1721,
for a Newtonian telescope btw.
The principle on which the parabolic mirror works was indeed known in antiquity.
However they did not know in antiquity how to make good mirrors, neither flat, not parabolic nor spherical (at least no evidence survived that they did know. The legend of Archimedes mirrors burning ships at a distance appears at a much later period. This is not possible even with modern mirrors).
This was a problem even in Newton's time, what metal to use. Good mirrors with silver coating on glass were produced much later. They also did not know in antiquity that one can use this principle to make telescopes. (Or, if they knew, this knowledge did not survive).
EDIT. History of (optical) telescopes is explained in great detail in the book
Telescopes, by Gerard P. Kuiper and Barbara M. Middlehurst, U. Chicago Press, 1961.
Very interesting reading. Includes the history of glass making and mirror polishing technology.