Buying a call and selling a put with the same strike price and expiry gives the same payments at expiry as a forward contract to buy the underlying asset at the strike price. Because the payments at expiry are the same, the values today are also the same, and the relationship follows from this. If the underlying asset pays dividends before expiry, the price of the underlying asset less the present value of those dividends is used.
Consider a hypothetical example in which the underlying price is 100, the strike price is 100 and the discount factor to expiry is 0.98. The call price is then 100 − 100 × 0.98 = 2 higher than the put price.
For American options, which can be exercised before expiry, the relationship becomes a pair of upper and lower bounds rather than an equality.