To determine if every number of $T$ is from $Y$, we need to analyze the given statements.

Statement I: The sum of any two numbers belonging to $T$ is even.

• If $T$ contains any odd number from $X$, then the sum of two odd numbers is even, which is consistent with the statement. However, if $T$ contains both odd and even numbers, the sum of an odd and an even number is odd, which contradicts the statement. Therefore, $T$ cannot contain any odd numbers, and all numbers in $T$ must be even, i.e., from $Y$.

Statement II: If both $p$ and $q$ are picked from $T$, then $(p - 1)q$ is even.

• If $p$ is odd, then $p - 1$ is even, making $(p - 1)q$ even regardless of whether $q$ is odd or even. If $p$ is even, then $(p - 1)$ is odd, and for $(p - 1)q$ to be even, $q$ must be even. This statement alone does not rule out the possibility of $T$ containing odd numbers, as it allows for $p$ to be odd and $q$ to be any number. Therefore, this statement alone is insufficient to conclude that all numbers in $T$ are from $Y$.

Thus, Statement I alone is sufficient to answer the question, while Statement II alone is not. Therefore, the correct option is A.