The primary limit on the speed of a parallel adder is the 'ripple propagation' of carries, since bit n cannot be computed until it is known whether or not there is a carry out of bit n-1. In a simple implementation, this effectively sequentially stacks the delays across the entire adder (consider the case of the case of the addition of 01 to 077777).
A number of 'carry lookahead' mechanisms have been devised to try and bypass the propagation delays inherent in sequential carries. (This has been the case since the days of Babbage, whose machines implemented clever carry mechanisms.)
(The adders in serial computers do not face this issue, of course.)