In parsimony analysis, the problem of inapplicables (see Maddison W.P. 1993. Syst. Biol. 42, 576-581) can be overcome by maximizing the amount of similarity that can be interpreted as homology, an idea that I first discussed in this 2002 talk.
Maximization of homology also provides the key to extend parsimony to the analysis of unaligned sequence data, as I discussed in this 2004 talk and in this 2005 paper. In that paper I showed that, in tree alignment programs such as POY, cost regime 3221 (gap opening cost three, transition and transversion costs two, and gap extension cost one) provides an optimal approximation for the cost set that maximizes homology when all instances of homology are equally weighted. A discussion of differential weighting of homologies can be found in this 2015 paper (section on approximations and section on sensitivity analysis).
Inapplicables as they arise in the classic approach are a special case of inapplicables as they arise in sequence data. This special case can be tackled with algorithms that are computationally less complex. A recent discussion can be found in the above 2015 paper (section on inapplicables). Anagallis is a computer program that provides tree searches with such algorithms. Release was originally announced by WHS XXXII (Rostock, 3-7 August 2013), but is postponed until the summer of 2016.