Webster and Purvis (2002) reported that the ML approach (Schluter et al. 1999) tends to give the most accurate results. However, that method assumes change is proportional to branch lengths (which you don't have). So using an assumption that change is speciational (i.e. parsimony, with branch lengths effectively equal), I think Butler and Losos (1997) examined that comparison and found that while neither is great, squared-change parsimony is less bad when trait evolution follows a Brownian Motion model. But as Martins (1999) points out, squared-change parsimony has the major downside of not providing any support index or confidence interval for values of reconstructed nodes, so you really don't know how much uncertainty is involved. It's also important to consider whether your trait displays a directional trend, as continuous ancestral state reconstruction methods tend to perform poorly under such a scenario (Oakley and Cunningham 2000).

Thanks for the references Nicholas. Is there a way by which one could discern directional trends in the data?

Also, am I right in thinking that ancestral state reconstruction by linear parsimony assumes a punctuated change model of evolution whereas square change parsimony assumes a gradual change model of evolution?

I'm not current on literature for your specific question, but why can't you use branch lengths from morphological data, if you have the dataset and not just the tree topology?