To determine if $x$ is even, we analyze the given statements:

Statement I: $x^2 y^2$ is even.

• If $x^2 y^2$ is even, then at least one of $x^2$ or $y^2$ must be even.
• If $x^2$ is even, then $x$ must be even (since the square of an odd number is odd).
• If $y^2$ is even, $y$ must be even, but this does not directly tell us about $x$.
• Therefore, this statement alone is not sufficient to determine if $x$ is even, as $y$ could be even while $x$ is odd.

Statement II: $1 + x^2 + y^2$ is odd.

• For $1 + x^2 + y^2$ to be odd, $x^2 + y^2$ must be even (since an odd number plus an even number is odd).
• For $x^2 + y^2$ to be even, both $x^2$ and $y^2$ must be even (since the sum of two odd numbers is even, but $x^2$ and $y^2$ being odd would make $x^2 + y^2$ even, contradicting the need for $x^2 + y^2$ to be even).
• Therefore, both $x$ and $y$ must be even for $x^2$ and $y^2$ to be even.
• This statement alone is sufficient to determine that $x$ is even.

Thus, the question can be answered using Statement II alone, but not Statement I alone. Therefore, the correct option is A.