Estimating parameters of Partial Differential Equations (PDEs) from noisy and indirect measurements requires solutions of ill-posed inverse problems. Such problems arise in a variety of applications such as geophysical, medical imaging, and nondestructive testing. These so called parameter estimation or inverse medium problems, are computationally intense since the underlying PDEs need to be solved numerous times until the reconstruction of the parameters is sufficiently accurate. Typically, the computational demand grows significantly when more measurements are available, which poses severe challenges to inversion algorithms as measurement devices become more powerful. In this paper we present jInv, a flexible framework and open source software that provides parallel algorithms for solving parameter estimation problems with many measurements. Being written in the expressive programing language Julia, jInv is portable, easy to understand and extend, cross- platform tested, and well-documented. It provides parallelization schemes that exploit the inherent structure in many parameter estimation problems and can be used to solve multiphysics inversion problems as is demonstrated using numerical experiments motivated by geophysical imaging.