This paper presents a convex approach for parameter estimation problems (PEPs) involving parameter-affine dynamic systems. By using the available state measurements, the nonconvex PEP is modified such that a convex approximation is obtained. The optimum delivered by this approximation is subsequently used to linearize the original PEP such that a refined solution is obtained. An assessment of the distance between the actual solution and the one delivered by the refined approximation is provided using perturbation analysis. Comparative results using a classical nonlinear parameter estimation procedure and the proposed approach are presented for two benchmark examples where the theoretical assessment of errors in solutions is corroborated numerically.