I want to assume I am in a general Black Scholes Model with $r=0$ and $\delta=0$ and the typical filtered probability space.
I know that $Call^{BS}(0, x, K, T) = Put^{BS}(0, K, x, T)$ with $x= S_0$, which is our start value, holds. I have proven this. My question is now how can I expand it to arbitrary $t \in [0, T]$ with $x = S_t$, i.e. $Call^{BS}(t, x, K, T) = Put^{BS}(t, K, x, T)$ with $x = S_t$?
It is for me of course somehow clear in a logical way but I have problems proving it. My idea was to use the markov property of our asset and get something like $Call^{BS}(t, x, K, T) = Call^{BS}(0, x, K, T) = Put^{BS}(0, K, x, T) = Put^{BS}(t, K, x, T)$ but this is of course not true yet. How do I change the starting value?
Thank you for your answers!