Put-call parity is nothing more than an equation that shows how the price of a (European) put option (on, say, a stock) relates to the price of a (European) call option (on the same stock). The equation arises from a simple finance fact:
A forward rate agreement (FRA) is essentially an agreement to enter into two loans (one long, one short) in the future: a fixed-rate loan and a floating-rate loan. (The difference between an FRA and an actual agreement to enter into these two loans is that the FRA will be settled at the beginning of the loan period, whereas the loans would be settled at the end). Specifically, an m × n FRA is an agreement to enter into two (n – m)-month loans starting m months from today.