Let $\{F(t, T), 0 \leq t \leq T\}$ be the forward process that satisfies an SDE of the form
\begin{align*}
dF(t, T) = \sigma F(t, T) dW_t,
\end{align*}
where $\sigma$ is the constant volatility, $\{W_t, t>0\}$ is a standard Brownian motion. The payoff at time $T_1$, where $0 < T_1 \leq T$, of a vanilla European forward option is of the form
\begin{align*}
\max(\psi (F(T_1, T)-K), \, 0),
\end{align*}
where $\psi = 1$, for a call option, and $-1$, for a put option. Note that, for any $0\leq t \leq T_1$,
\begin{align*}
F(T_1, T) = F(t, T) \exp\Big(-\frac{\sigma^2}{2} (T_1 -t) + \sigma \sqrt{T_1 -t} \xi \Big),
\end{align*}
where $\xi$ is a standard normal random variable. Then the value at time $t$ of the option payoff above is given by
\begin{align*}
d(t, T_1)\psi\Big[F(t, T) \Phi\big(\psi d_1(F)\big) -K \Phi\big(\psi d_2(F)\big) \Big],
\end{align*}
where $d(t, T_1)$ is the discount factor,
\begin{align*}
d_1 (F) = \frac{\ln \frac{F(t, T)}{K} + \frac{\sigma^2}{2} (T_1 -t)}{\sqrt{T_1-t}\,\sigma},
\end{align*}
and
\begin{align*}
d_2 (F) = \frac{\ln \frac{F(t, T)}{K} - \frac{\sigma^2}{2} (T_1 -t)}{\sqrt{T_1-t}\,\sigma}.
\end{align*}
That is,
\begin{align*}
C(F, t) = d(t, T_1)\Big[F(t, T) \Phi\big(d_1(F)\big) -K \Phi\big(d_2(F)\big) \Big],
\end{align*}
and
\begin{align*}
P(F, t) = d(t, T_1)\Big[K \Phi\big(-d_2(F)\big) -F(t, T) \Phi\big(-d_1(F)\big)\Big],
\end{align*}
Note that, by replacing $F$ in $d_1$ with $K^2/F(t, T)$,
\begin{align*}
d_1 \Big(\frac{K^2}{F}\Big) &= \frac{\ln \frac{K^2/F(t, T)}{K} +\frac{\sigma^2}{2} (T_1 -t)}{\sqrt{T_1-t}\,\sigma}\\
&= \frac{-\ln \frac{F(t, T)}{K} + \frac{\sigma^2}{2} (T_1 -t)}{\sqrt{T_1-t}\,\sigma}\\
&= -d_2(F).
\end{align*}
Similarly,
\begin{align*}
d_2 \Big(\frac{K^2}{F}\Big) &= \frac{\ln \frac{K^2/F(t, T)}{K} -\frac{\sigma^2}{2} (T_1 -t)}{\sqrt{T_1-t}\,\sigma}\\
&= \frac{-\ln \frac{F(t, T)}{K} - \frac{\sigma^2}{2} (T_1 -t)}{\sqrt{T_1-t}\,\sigma}\\
&= -d_1(F).
\end{align*}
Then
\begin{align*}
\frac{F}{K}P\bigg(\frac{K^2}{F}, t \bigg) &= d(t, T_1)\frac{F}{K}\Bigg[K \Phi\bigg(-d_2\bigg(\frac{K^2}{F}\bigg)\bigg) -\frac{K^2}{F} \Phi\bigg(-d_1\bigg(\frac{K^2}{F}\bigg)\bigg)\Bigg]\\
&= d(t, T_1)\bigg[F \Phi\Bigg(-d_2\bigg(\frac{K^2}{F}\bigg)\bigg) -K \Phi\bigg(-d_1\bigg(\frac{K^2}{F}\bigg)\bigg)\Bigg]\\
&= d(t, T_1)\Big[F(t, T) \Phi\big(d_1(F)\big) -K \Phi\big(d_2(F)\big) \Big]\\
&= C(F, t).
\end{align*}
