We deduce a procedure to use balanced truncation for parameter-dependent differential-algebraic systems. For this, we have to solve parameter-dependent Lyapunov equations. Since solving large-scale Lyapunov equations for every parameter leads to very high computational costs we utilize the reduced-basis method to get a reduced Lyapunov equation that we can solve instead. In order to use this method, we first make the matrix pencil of the system strictly dissipative to apply error estimators in the reduced basis method. Additionally, we deal with the algebraic parts of the system by using projections that lead to ordinary differential equations.