I dno't really know what is the issue, but it seems that this code is the culprit:

\mathopen{}\mathclose\bgroup ##1 ##2 \aftergroup\egroup ##3
and the parameters are {left(}{expression}{\right)}

The problem is that it is declared as macro at the time of calling \DeclarePairedDelimiter, which is in the preamble in your case. As configurations from .4ht files are included after the preamble, we probably can't configure that. One solution which is possible is to use alternative4ht package from helpers4ht bundle[1], for which I provided some configurations[2].

Modified file which works for me:

\documentclass{article}
\usepackage{alternative4ht}
\altusepackage{mathtools}
\makeatletter
\makeatother
\DeclarePairedDelimiter{\paren}{(}{)}
\begin{document}
Examples:

eq1: $c^{2} \paren{a^{2}+b}$ !

eq2: $c^{2} \paren*{a^{2}+b}$ !
\end{document}

I am curious that there aren't any way to provide configurations for commands in the preamble, at least I don't know about any :(

[1] https://github.com/michal-h21/helpers4ht
[2] https://github.com/michal-h21/helpers4ht/blob/master/alt-packages/mathtools-alt4ht.sty