Some time ago, someone ask me about which makes cladistics different from other phylogenetic approaches. For most people the answer is straightforward: a cladist is the one that uses parsimony (e.g. ).
I think parsimony is very important for cladists, but I do not think that you are only a cladists when use parsimony. There is a lot of papers from author that use parsimony, but which I'm not think that are cladistic papers. Also there are a lot of works form author that do not use parsimony, but they are clearly under the cladistic framework (I think on the old one authors: Hennig, Brundin, Wydodzinsky, etc., all of them are cladists, but no one uses parsimony, at least in an explicit [=numerical] way).
So, if it is not an algorithm, where is the difference? When are you a cladists?
A key part of cladistic thinking is monophyly, but monophyly actually is usually understood as a topological term (i.e. just the relationships depicted on a tree), but I think this passage from Hennig  is clear:
“The supposition that two or more species are more closely related to one another than to any other species, and that, together they form a monophyletic group, can only be confirmed by demonstrating their common possession of derivative characters (“synapomorphy”). When such character have been demonstrated, then the supposition has been confirmed that they have been inherited from an ancestral species common only to the species showing these characters.”
Then, for a cladist, monophyly is not just about the relationship, but the characters that support such relationship. When a cladist answer the question “Is this group monophyletic?” he/she not only shows a tree, he/she also shows the characters that support such grouping.
Then cladistics is not about the particular algorithm used to infer the relationships, but about the characters that support such groups. It is not estrange then that most cladists used morphology in their analyzes. But even, using just molecular data, you can be a cladist, when referring to each clade you are also discussing the synapomorphies of that clade (either molecular and/or morphological). Then also, you might prefer some particular algorithm over another.
Here is an example, there are two papers, one by Wiens et al.  about the phylogeny of Squamates, and other from Beutel et al.  on the phylogeny of Holometabola. Both papers use parsimony and bayesian analysis on their data, and both seems to prefer the tree resulting from the bayesian analysis. Both are about the same size (in pages), and discuss highl level relationships of a largely diverse group using both molecular and morphological data sets. One of this works is clearly a cladistic work, and the other not.
If you look to Beutel et al. paper, you will find a lot of discussion about the characters that support each clade. For them, support is not just the Bremer' support value, jackknife frequency or Bayesian probability, they are important of course, but they are meaningless without an explicit reference to the characters on the node. So, even if they prefer Bayesian analysis, they continue to be cladistis.
On the other hand, you have de Wiens et al. paper. For that authors, just the relationships are important, they discuss groupings and bayesian probabilies, but no discussion on any explicit character for any group is made. Then, even in the case of Wiens et al. only use the results of parsimony, they are not cladists: characters are not important for them.
A cladist is not the one which write a paper with the more up-to-date search strategies, or support measures, or larges amounts of new molecular data, or parsimony analysis. All of that things are important. But in a cladistic paper, the important think are the characters that support the phylogeny. A good cladistic paper is a paper about the characters.
 Felsenstein, J. 2001. The troubled growth of statistical phylogenetics. Syst. Biol. 50: 465-467. Doi: 10.1080/10635150119297 [available free at Syst. Biol. site]
 Williams, D.M., Ebach, M.C. 2007. Foundations of systematics and biogeography. Springer, New York (USA).
 Hennig, W. 1965. Phylogenetic systematics. Ann. Rev. Entomol. 10: 97-116. Doi: 10.1146/annurev.en.10.010165.000525 [available free here]
 Wiens, J.J. et al. 2010. Combining phylogenomics and fossils in higher-level squamate reptile phylogeny: molecular data change the placement of fossil taxa. Syst. Biol. 59: 674-658. doi: 10.1093/sysbio/syq048 [available free at Syst. Biol. site]
 Beutel, R.G. et al. In press. Morphological and molecular evidence converge upon a robust phylogeny of the megadiverse Holometabola. Cladistics. Doi:10.1111/j.1096-0031.2010.00338.x [available free at Cladistics site]