Well, clearly each statement alone is insufficient. You either do not have even/oddness of x or y, and that means regardless of what the other two values are, you can change y or x (for statements (1) and (2) respectively) and you can get a different answer.
Note that what the statement is essentially saying, is that for x and z has to be both even or both odd (for statement 1), as otherwise they cannot add together to be even. The same goes for statement 2.
So when we combine the two statements, we can start testing different situations. If x is odd, then z is odd, and thus y must also be odd. We then have
x + y + 2z be odd + odd + 2x odd, which comes out to be even. If x is even, then z is even, and thus y must also be even. We then have
x + y + 2z be even + even + 2x even, which still comes out to be even. As such, both statements together are sufficient, always giving us an even answer.
Hence C. _________________
Not a professional entity or a quant/verbal expert or anything. So take my answers with a grain of salt.