Is it possible to reverse Diffie-Hellman key exchange? Is it possible to reverse Diffie-Hellman key exchange to get the private key of one of the parties, given a large prime number 'p' .
It is certainly possible given enough compute resources.

However, if you mean to ask whether it is computationally feasible for an eavesdropper to derive the private keys, then- no, it is not.

Refer to the secrecy chart on Wikipedia: 

The private keys a and b are meant to be known only to Alice and Bob respectively.

In the real world, the Logjam attack is one example of how adversaries with nation state-level compute resources (such as the NSA) could theoretically break Diffie Hellman implementations, relying on the fact that a small set of primes were used across most common implementations.