We study weighted norm inequalities for families of operators depending on a parameter, $\varepsilon=\{\varepsilon_Q\}_{Q\in\mathcal{D}}$, representing a sequence of real numbers indexed by a dyadic system $\mathcal{D}$ in $\mathbb{R}^n$. We give necessary and sufficient conditions describing the weights for which such operators satisfy the corresponding weighted strong-type and weak-type bounds. Our results use a more general modification of the classical Muckenhoupt $A_p$ condition involving the parameter $\varepsilon$.