A parallel current can destabilize a single flux line (FL) or an array of FLs. We consider the effects of pinning by point impurities on this instability. The presence of impurities destroys the long-range order of a flux lattice, leading to the so-called Bragg glass (BrG) phase. We first show that the long-range topological order of the BrG is also destroyed by a parallel current. Nonetheless, some degree of short-range order should remain, whose destruction by thermal and impurity fluctuations, as well as the current, is studied here. To this end, we employ a cage model for a single FL in the presence of impurities and current, and study it analytically (by replica variational methods) and numerically (using a transfer matrix technique). The results are in good agreement, and in conjunction with a Lindemann criterion, provide the boundary in the magnetic-field--temperature plane for destruction of short-range order. In all cases, we find that the addition of impurities or current (singly or in combination) leads to a further increase in equilibrium FL fluctuations. Thus pinning to point impurities does not stabilize FLs in a parallel current ${j}_{z},$ although the onset of this instability is much delayed due to large potential barriers that diverge as ${j}_{z}^{\ensuremath{-}\ensuremath{\mu}}.$