= Solution
For European options with the same strike and maturity, the terminal payoff identity is
$$
(S_T-X)^+-(X-S_T)^+=S_T-X.
$$
Thus long the call and short the put has the same terminal payoff as one <stock> less $X$ units of the maturity-$T$ <zero-coupon bond>. With no dividends, the <stock> strategy has no intervening payments. The <law of one price> gives <put-call parity>:
$$
\boxed{C_E(t)-P_E(t)=S_t-XB(t,T).}
$$
Equivalently the two nonnegative terminal-payoff <portfolios>, call plus strike bond and put plus <stock>, agree in every state. The parity proof does not require lognormal <stock> prices or positive interest.
For <American options> the exercise times can differ, so equality of their maturity payoffs is not an identity between their full rights. Under the nonnegative-interest, non-dividend assumptions of part (a), $C_A=C_E$ while $P_A=P_E+\Pi_P$ with early-exercise premium $\Pi_P\ge0$. Hence
$$
C_A-P_A=S_t-XB(t,T)-\Pi_P,
$$
which generally differs from European parity. For a concrete limiting state, if the <stock> is zero and remains zero and interest is positive, then $C_A=C_E=0$, $P_A=X$ and $P_E=XB(t,T)<X$. \b[European parity is not a general American-option equality.] It holds in a particular case only when the relevant early-exercise premiums cancel, or both vanish.
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