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Bunuel
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Given: X & Y are integers

Assumptions required: X & Y can be +ve, -ve, or “0”.

Statement 1 : y^(2x) + 8 = odd
Above statements imply -> odd + even = odd
Therefore : y^(2x) = odd
Now, if X is zero then y^(2x) will always be equal to "one" i.e. odd, no matter whether y is odd or even, y^(2x) will always result in odd i.e. "1" hence, Y's Odd-even position can’t be arrived.
Insufficient

Statement 2: (y^7) X = odd
i.e. odd x odd = odd
=> (y^7) = odd
Therefore y = odd (note: an odd number multiplied by itself will always be an odd number)
Sufficient

Correct choice = B
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Statement 1:
x=0

y could be odd or even

Not sufficient

Statement 2:
y must be odd

Sufficient

B
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