All I can point at is from the array language world, with the notable thing of Co-dfns from Aaron Hsu, an APL compiler capable of running on the GPU (pointless as that may be): https://github.com/Co-dfns/Co-dfns/tree/master; and a 284-page dissertation: https://scholarworks.iu.edu/dspace/items/3ab772c9-92c9-4f59-...
And Marshall Lochbaum following (https://mlochbaum.github.io/BQN/implementation/codfns.html) that work, and covering some basics of parsing expressions of infix & prefix ops to stack-based/RPN at the bottom of https://mlochbaum.github.io/BQN/implementation/index.html, though the code is in BQN.
From that same author, parsers written in BQN, that I think should have sub-linear critical paths:
Markdown: https://github.com/mlochbaum/BQN/blob/master/md.bqn
XML: https://github.com/mlochbaum/bqn-libs/blob/master/xml.bqn
and a compiler of BQN itself: https://github.com/mlochbaum/BQN/blob/master/src/c.bqn (outputs stack-based bytecode).
Given how leniently html deals with errors and that some of the parsing rules have some context sensitivity (the various modes) actually exploiting large-scale parallelism is probably a very difficult bit of engineering. Additionally, there may be corner cases that have much more context dependence than normal. Also, parsing is already probably not a huge part of a browsers runtime, and aggressive parallelism might be faster but more power hungry for a very small latency win. But I can't think of a specific fundamental limitation that prevents the normal parsing case from being quite parallelizable; I believe it's "just" an engineering problem.