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MATL, 6 bytes

toYs~)

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Explanation

t % Input array implicitly. Duplicate
o % Parity of each entry. Gives 0 for even entries, 1 for odd
Ys % Cumulative sum
~ % Logical negate. Gives true for the first run of even entries, and then false
) % Use this as logical index into the original array. Implicitly display