Let V be the vertex of a parabola, F its focus, X a point on its symmetry axis, and A a point on the parabola such that AX is orthogonal to VX. It was well within the power of the Greeks to prove relations such as $VX:XA = XA:4VF$. If you introduce coordinate axes, set $x = VX$, $y = XA$ and $p = VF$, you get $y^2 = 4px$, the modern form of the equation of a parabola.
Everything now depends on what "invention of coordinate geometry" means to you. I do not think that the Greeks' work on conics should be confused with coordinate geometry since they did not regard the lengths occurring above as coordinates. It's just that parts of their results are very easily translated into modern language.
In a similar vein, Eudoxos and Archimedes already were close to modern ideas behind integration, but they did not invent calculus. Euclid, despite Heath's claim to the contrary, did not state and prove unique factorization. And Euler, although he knew the product formula for sums of four squares, did not invent quaternions (Blaschke once claimed he did).
In any case, we are much more careful now with sweeping claims such as "Appolonius knew coordinate geometry" than historians were, say, 100 years ago.
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