  We consider a general class of SPDEs in $\mathbb{R}^d$ driven by a Gaussian
spatially homogeneous noise which is white in time. We provide sufficient
conditions on the coefficients and the spectral measure associated to the noise
ensuring that the density of the corresponding mild solution admits an upper
estimate of Gaussian type. The proof is based on the formula for the density
arising from the integration-by-parts formula of the Malliavin calculus. Our
result applies to the stochastic heat equation with any space dimension and the
stochastic wave equation with $d\in \{1,2,3\}$. In these particular cases, the
condition on the spectral measure turns out to be optimal.
