We study cells with respect to the $p$-canonical basis of the Hecke algebra of a crystallographic Coxeter system (see arXiv:1510.01556, arXiv:1901.02323) and their compatibility with standard parabolic subgroups. We show that after induction to the surrounding bigger Coxeter group the cell module of a right $p$-cell in a standard parabolic subgroup decomposes as a direct sum of cell modules. Along the way, we state some new positivity properties of the $p$-canonical basis.