Reach set computations are of fundamental importance in control theory. We consider the reach set problem for open-loop systems described by parametric inhomogeneous linear differential systems and use real quantifier elimination methods to get exact and approximate solutions. The method employs a reduction of the forward and backward reach set and control parameter set problems to the transcendental implicitization problems for the components of special solutions of simpler non-parametric systems. For simple elementary functions we give an exact calculation of the cases where exact semialgebraic transcendental implicitization is possible. For the negative cases we provide approximate alternating using discrete point checking or safe estimations of reach sets and control parameter sets. Examples are computed using the redlog and qepcad packages.