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 it is shown 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. I announced release at WHS XXXII (Rostock, 3-7 August 2013) but afterwards decided to postpone release until the program could guarantee optimality of tree scores obtained under a much wider range of conditions than initially announced (optimality of reported tree scores is now guaranteed as long as no independent regular Fitch or Farris optimization of a particular feature has a solution where an individual region of absence has more than eleven neighbouring regions of presence; looks like a reasonable enough assumption in practice). The full presentation of that talk is available at ResearchGate here. The related presentation of my 2012 Riverside WSH XXXI talk can be found here. I intend to release a beta of the program before the end of the year.

*Update 1 November 2017. *Last Wednessday Brazeau, Guillerme and Smith published this interesting paper on morphological analysis with inapplicable data on BioRXiv.
The main difference with my approach seems to be that they independently optimize single-column characters with inapplicables rather than character hierarchies as a whole. This may give good results under a wide range of conditions, but in general the optimization of a character hierarchy on a tree cannot be reduced to a series of independent single-character optimizations on that tree.
Doing so may yield a fast approximation for the score of a character hierarchy on a tree, but it can miss optimal state reconstructions, miss the optimal score, and ultimately identify non-optimal trees as optimal during tree search.
The first of these three issues can be illustrated using the example of their Fig. 3. I'll discuss it here in some more detail soon.