Why Must the Options in the Parity Equation Be European-Style?

The options in the standard put-call parity equation must be European-style because they can only be exercised on the expiration date. This fixed exercise date simplifies the valuation and allows for a clean, deterministic relationship between the call, put, and forward contract.

American-style options, which can be exercised at any time, introduce the complexity of early exercise premium, which breaks the simple mathematical relationship of the standard parity formula.