What kind of parser does FASM employ?

What kinds are there?

Top-down, bottom-up, shift-reducing, I don't know all..

Well, it takes all symbols/numbers/whatever one by one and translates them to internal "tokens" which are easier to work with than text. Not sure where that stands in your scheme.

What you described is the job of lexical analyser, and in fasm the lexical analyser is integrated into preprocessor. Parser is the next stage and is a little bit more complex. And fasm's parser is abit specific one, because it only recognizes some general constructs, leaving some additional analysis of operand structures to still be done later by assembler module.

I would believe that the assembly instructions don't need a powerful parser, but evaluating expressions and perhaps macros would call for a more powerful recursive parser, perhaps a shift-reducing one.

I'm thinking of writing an 6502 assembler for my thesis, that's my I'm wondering.

Evaluating macros is in fasm done by preprocessor, so it is in fact another separate parser, since fasm's preprocessor language is completely separate from the assembly. Also, fasm's preprocessor is generally top-down with only minor exceptions (like "rept" directive expression calculator).
As for the main parser of fasm, it is more complex, and could be perhaps characterized as a mix of top-down and bottom-up approaches if you tried to categorize it in this way. But I feel that this terminology only suits well the general theoretical cases, and not a highly specialized one, like the fasm's parser. In fact, fasm's parser module is just a translator, which translates the preprocessed source (which is already tokenized) into a language of assembler module, which is then interpreted in multiple passes by assembler module (and some of the syntax checking and analysis still is left to be done upon interpretation by assembler, parser just translates it regardless of it being nonsense or not).