Solution: The equation $x^2 - y^2 = 2025$ factors as $(x - y)(x + y) = 2025$. Since $x$ and $y$ are integers, both $x - y$ and $x + y$ must be integers. Let $a = x - y$ and $b = x + y$, so $ab = 2025$. Then $x = \fraca + b2$ and $y = \fracb - a2$. For $x$ and $y$ to be integers, $a + b$ and $b - a$ must both be even, so $a$ and $b$ must have the same parity. - Crosslake