Context free grammar for language with even number of $0$'s and $1$'s
The most common approach to show that two sets are the same is to show that they are subsets of each other.

In this case, that means (1) every string with an even number of $0$s and $1$s has a derivation in the grammar, and (2) every derivation of the grammar has an even number of $0$s and $1$s.

In attempting (2), a reasonable approach is to prove inductively, from smaller strings to bigger strings.

For example, considering $S \to 0S0S$, since the number of $0$s increases by 2 and the number of $1$s stays the same, the resulting string has the desired property when the smaller $S$ strings in the right hand side do.

However, I think you'll find the $S \to BABAS$ case hard to prove correct, which suggests that's where you need to look for a counterexample. Indeed, here's one:

0101010 = (B:0)(A:1)(B:0)(A:1010)(S:) = (B:0)(A:1)(B:0)(A:(1(A:0)1(A:0)))(S:)