American call and put option relationship

Question

If an American call option and an American put option both have the same strike price K and expire at time T, with no dividends, how can we find a relationship between their prices? I get that if they are both held till expiry, we can use the put-call parity for European options, but if they aren't, how can we use the arbitrage argument to find the relationship between their prices? I am not sure why they will not have the same price