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Is it possible to create a parser for pure LaTeX (no plain TeX, no TeX primitive) without using any TeX engine that supports total LaTeX?
I know about an iOS app to create and typeset LaTeX and I don't think that they are using a TeX engine.
Is there such a Parser?

If you write a parser you can define the subset of latex that you support. (There isn't really a useful definition of "Pure LaTeX with no primitives".)

For instance MathJax has a parser for a subset of LaTeX math markup, written in JavaScript, and LaTeXML has a parser for almost complete TeX written in perl, which does not include any TeX execution. LaTeXML's parser is perhaps the closest to what you ask, as far as I understand the question. https://github.com/brucemiller/LaTeXML

Here is an example that only uses commands defined in core latex (the shortvrb package is part of the base LaTeX2e release, so it is as fundamental art of latex as say section which is defined in article class from the same base release files

documentclassarticle
usepackageshortvrb


begindocument

MakeShortVerb*

 bfseries ** some text

DeleteShortVerb*

 bfseries ** some text

enddocument

Note that it is not possible to statically assign any tokenisation to ** in the first case it produces the two character tokens { in the second case it produces two character tokens ** (the first one being bold).

It would be reasonable to produce a LaTeX parser for a subset of the language that did not include this kind of construct, but you need to define the subset it isn't enough to say "not plain TeX or primitives" there are plain constructs that can be easily parsed, and there are LaTeX constructions that can not be parsed in general without access to a full tex typesetting system.

I think this already occurs for document conversion software such as pandoc, and others on the internet. Generally speaking these converters only parse a subset of the commands. In addition regex can be used to extract certain tags of interest.