In this work, we have studied the accretion of dark energies onto a Morris-Thorne wormhole. Previously, in ref. (González-Díaz, arXiv:hep-th/0607137), it was shown that for quintessence like dark energy, the mass of the wormhole decreases, and for phantom like dark energy, the mass of the wormhole increases. We have assumed two types of dark energy: the variable modified Chaplygin gas and the generalized cosmic Chaplygin gas. We have found the expression of the wormhole mass in both cases. We have found the mass of the wormhole at late universe and this is finite. For our choices of the parameters and the function B(a), these models generate only quintessence dark energy (not phantom) and so the wormhole mass decreases during the evolution of the universe. Next we have assumed the five kinds of parametrizations of well-known dark-energy models. These models generate both quintessence and phantom scenarios i.e., phantom crossing models. So if these dark energies accrete onto the wormhole, then for the quintessence stage, the wormhole mass decreases up to a certain value (a finite value) and then again increases to an infinite value for the phantom stage during whole evolution of the universe. That means that if the five kinds of DE accrete onto a wormhole, the mass of the wormhole decreases up to a certain finite value and then increases in the late stage of the evolution of the universe. We have also shown these results graphically.