With number property questions such as these, I "plug in" even and odd into the formulas.

odd*odd = odd, odd*even = even, even*even = even
and odd+odd = even, odd+even = odd, and even+even = even

e = even, o = odd

A)
x is even: $e^2 + 6(e) + 9 = e + e + o = o$
x is odd: $o^2 + 6(o) + 9 = o + e + o = e$

B)
x is even: $e^2 + 7(e) + 12 = e + e + e = e$
x is odd: $o^2 + 7(o) + 12 = o + o + e = e$

C)
x is even: $e^2 + 4(e) - 21 = e + e - o = o$
x is odd: $o^2 + 4(o) - 21 = o + e - o = e$

D)
x is even: $e^4 + 4(e) - 12 = e + e - e = e$
x is odd: $o^4 + 4(o) - 12 = o + e - e = o$

E)
x is even: $e^2 + 8(e) + 15 = e + e + o = o$
x is odd: $o^2 + 8(o) + 15 = o + e + o = e$