Yes, and there are also already multiple equal-area projections with similar properties, too. For example, the Goode homolosine "orange-peel" projection only loses out by failing to show contiguous oceans (emphasizing land masses instead) and having a discontinuity where two simpler projections are joined. It gives equal-area projection and improved shapes (as compared to fully homolographic projections) while still showing "curved sides [that] suggest the spherical form of Earth" (arguably, far more so). Its parallels are straight, and furthermore they are equidistant in the central latitudes (in which it's based on a sinusoidal projection).

Oh, and it was developed over a century ago, and already in common use when Arno Peters started his activism for the Gall-Peters projection (called this even though Peters made no refinements in independently developing a projection identical to Gall's 1855 work, and even initially mis-described it).