We can rule out A,B because Sports Medicine Programs is plural. We need "are".

In C, "their" can point to either athletes or Sports Medicine Programs.

In D, too wordy and no need to have toward ... and toward .... enhancement of atheletes' performance is not needed

E suits the bill.

In C, it should be "athletes' " instead of "athletes".
D is too wordy.
E is the best option.

The Sports Medicine Programs of the Olympic Training Center, a complex where final try outs are held for athletes representing the US in the Olympics, is geared toward enhancing athletes' performance and toward their preparation for international competition.

A.) is geared toward enhancing athletes' performance and toward their preparation --> is is singular --Programs is plural
B.) is geared to enhance the performance of athletes and to prepare them --> is is singular --Programs is plural
C.) are geared to enhance athletes performance and their preparation ---> Their --> programs or athletes
D.) are geared toward the enhancement of athletes' performance and toward preparing them --> them is ambiguous
E.) are geared toward enhancing the performance of athletes and preparing them --> cream of the crap...best one

Parallelism on E
geared toward: ENHANCING AND PREPARING

Can you please reveal the source...

I find Op C much better than Op E in terms of meaning clarity except that ambiguous THEIR, I have seen some questions(people posted) from gmat prep where pronoun ambiguity is considered secondary issue...parallelism is there in Op C also...may be if some expert suggest something else which is wrong in Op C...please reveal the source..??

There may be some engineering contexts where geared to is a correct idiom (I don't know, I'm not an engineer), but the expected idiom is geared toward.