  In this paper, we study linear backward parabolic SPDEs and present new a
priori estimates for their weak solutions. Inspired by the seminal work of Y.
Hu, J. Ma and J. Yong from 2002 on strong solutions, we establish
$L^p$-estimates requiring minimal assumptions on the regularity of the
coefficients, the terminal data, and the external force. To this end, we derive
a new It\^{o}'s formula for the $L^p$-norm of the solution, extending the
classical result in the $L^2$-setting. This formula is then used to improve
further the regularity of the first component of the solution up to $L^\infty$.
Additionally, we present applications such as the controllability of backward
SPDEs with $L^p$-controls and a local existence result for a semilinear
equation without imposing any growth condition on the nonlinear term.
