The issue comes from \AddLayersToPageStyle{@everystyle@}{chapterthumb}. It adds layer chapterthumb to style @everystyle@. Therefore the chapterthumbs are used with all layer page styles: empty, scrheadings, plain (alias of plain.scrheadings), ...
You can define a new page style for the part pages and add chapterthumbs only to page styles scrheadigs and plain.scrheadings:
\documentclass[oneside]{scrbook}
\usepackage{fontawesome}
\usepackage[french]{babel}
\usepackage{chapterthumb}% http://mirrors.ctan.org/info/examples/KOMA-Script-6/Anhang-B/source/chapterthumb.sty
\usepackage{csquotes}
\usepackage[T1]{fontenc}
\usepackage{lipsum}
\usepackage{scrlayer-scrpage}
\newpairofpagestyles[scrheadings]{part}{}
\renewcommand\partpagestyle{plain.part}
\AddLayersToPageStyle{scrheadings}{chapterthumb}
\AddLayersToPageStyle{plain.scrheadings}{chapterthumb}
\addtokomafont{chapterthumb}{\bfseries}
\usepackage{hyperref}
\begin{document}
\part{Foo}
\chapter{Pas du tout}
\lipsum[1]
\chapter{Un peu}
\lipsum[2]
\part{Bar}
\chapter{Beaucoup}
\lipsum[3]
\chapter{Passionnément}
\lipsum[4]
\end{document}
Result:

Or you can remove layer chapterthumb from @everystyle@ on part pages:
\documentclass[oneside]{scrbook}
\usepackage{fontawesome}
\usepackage[french]{babel}
\usepackage{chapterthumb}% http://mirrors.ctan.org/info/examples/KOMA-Script-6/Anhang-B/source/chapterthumb.sty
\usepackage{csquotes}
\usepackage[T1]{fontenc}
\usepackage{lipsum}
\usepackage{scrlayer-scrpage}
\AddLayersToPageStyle{@everystyle@}{chapterthumb}
\addtokomafont{chapterthumb}{\bfseries}
\newcommand*{\oldpartheadendvskip}{}
\let\oldpartheadendvskip\partheadendvskip
\renewcommand\partheadendvskip{%
\RemoveLayersFromPageStyle{@everystyle@}{chapterthumb}%
\oldpartheadendvskip
\AddLayersToPageStyle{@everystyle@}{chapterthumb}%
}
\usepackage{hyperref}
\begin{document}
\part{Foo}
\chapter{Pas du tout}
\lipsum[1]
\chapter{Un peu}
\lipsum[2]
\part{Bar}
\chapter{Beaucoup}
\lipsum[3]
\chapter{Passionnément}
\lipsum[4]
\end{document}
\pagestyle{empty}before\part{Bar}. It changes the page style of the normal (where no chapter or part starts) pages to empty. And it changes the page style of the page before part »Bar« starts. – esdd Sep 25 '19 at 20:09