Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

When p, m, and n are integers, is p an even?

1) m+n=even
2) p+m=even


In the original condition, there are 3 variables, which should match with the number of equations. So you need 3 equations. For 1) 1 equation, for 2) 1 equation, which is likely to make E the answer.
When 1) & 2), m=n=p=2 -> yes, m=n=p=1 -> no, which is not sufficient.
Thus, E is the answer.