  In this paper, we improve the error term in the prime geodesic theorem for
the Picard manifold $ \mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathbb{H}^3
$. Instead of $ \mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathbb{H}^3 $, we
establish a spectral large sieve inequality for symmetric squares over
$\mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathrm{PSL}_2 (\mathbb{C}) $.
This enables us to improve the bound $ O (T^{3+2/3+\varepsilon}) $ of Balkanova
and Frolenkov into $ O (T^{3+1/2+\varepsilon}) $ for the second moment of
symmetric square $L$-functions over $ \mathrm{PSL}_2 (\mathbb{Z} {[i]})
\backslash \mathbb{H}^3 $. The basic idea is to enlarge the spherical family
$\Pi_c^{0} (T)$ of Maass cusp forms on $ \mathrm{PSL}_2 (\mathbb{Z} {[i]})
\backslash \mathbb{H}^3 $ into the family $ \Pi_c (T, \sqrt{T}) $ of cuspidal
representations on $ \mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash
\mathrm{PSL}_2 (\mathbb{C}) $.
