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We give a sufficient condition for the strict parabolic power concavity of the convolution in space variable of a function defined on $$\mathbb {R}^n \times (0,+\infty )$$ and a function defined on $$\mathbb {R}^n$$ . Since the strict parabolic power concavity of a function defined on $$\mathbb {R}^n \times (0,+\infty )$$ naturally implies the strict power concavity of a function defined on $$\mathbb {R}^n$$ , our sufficient condition implies the strict power concavity of the convolution of two functions defined on $$\mathbb {R}^n$$ . As applications, we show the strict parabolic power concavity and strict power concavity in space variable of the Gauss–Weierstrass integral and the Poisson integral for the upper half-space.