Which algorithms can be expressed using a total functional language with data parallel operators?
You might want to look at the old programming language NESL which took these ideas seriously. The language is based on operations on collections, essentially lists and lists of lists and so forth, but I think trees and graphs were also considered, via tricky encodings. Some of the research done in that project (in the mid 1990s) could help answer your question. The PhD thesis (available as a book) Guy E. Blelloch. Vector Models for Data-Parallel Computing. MIT Press, 1990 may provide some answers. It was some time ago since I looked at it.

Work done on the Bird-Meertens Formalism (BMF) falls into this category too, as did the language Charity. From the Charity wikipedia page it says that the language is not Turing complete, but can express Ackermann's function, which means that it's more than primitive recursive. Both BMF and Charity involve operators like folds and scans (catamorphisms, anamorphisms, etc), and they have their roots in Category Theory.

I short, imprecise answer is that you can express quite a lot.