EDIT
The current solution's interaction with reledmac is far from ideal.
It works nicely if no critical notes are needed, f.i., if reledmac is used only to have line numbering. (Which might be the focus of the original question, since no mention was made of footnotes.)
It also works with endnotes, but only if the custom tcolorbox is enclosed within the critical paragraph:
\pstart
\begin{criticalmarginsbox}
<Text>
\end{criticalmarginsbox}
\pend
Finally, the solution does not work with critical footnotes properly: if the critical paragraph is enclosed within the custom tcolorbox, the footnotes are lost. If, on the other hand, the box is within the paragraph, the footnotes are realized correctly, but the line numbering in the margin is lost.
The original answer
Let's assume that your blocks of critical text are to have a larger inner and a smaller outer margin, 4em and 1em respectively.
Setting parity-dependent page margins can be easily done globally (f.i. with pkg geometry).
On the other hand, setting such margins on a paragraph basis is, in general, an open problem. This is because (to quote @Werner) TeX sets a complete paragraph at a time and doesn't take any account of page breaks and possible layout changes within a paragraph.
Each single paragraph will use the same lengths, in the same absolute order, that were established at the start: it doesn't switch left with right depending on page parity.
Even a brute-force approach, like setting the margins via low-level macros at each paragraph start, is hopeless:
\leftskip=4em
\rightskip=1em
\pstart <text> \pend
\leftskip=1em
\rightskip=4em
\pstart <text> \pend

Fortunately, this answer suggests a satisfactory automatic solution.
We exploit the functions of tcolorbox, specifically its ability to toggle properties based on page parity, and define a \tcolorbox without colors with the desired margins.
To take care of the indentation, we specify the option before upper={\parindent=0pt}, i.e., have the code \parindent=0pt issued every time a criticalmarginsbox begins.
tcolorbox adds some vertical spacing around the box. You might find this desirable, as it makes the critical blocks stand out. Otherwise, keys such as before skip balanced and friends can be used to suppress this, as in my example.
\documentclass[foolscap, 11pt]{octavo}
\usepackage[marginparwidth=40pt, headsep=20pt,showframe]{geometry}
\usepackage{fontspec}
\usepackage{ebgaramond}
\usepackage{blindtext}
\usepackage{reledmac}
\usepackage{tcolorbox}
\tcbuselibrary{breakable}
\newtcolorbox{criticalmarginsbox}{
parbox=false,
left skip=4em,
right skip=1em,
breakable,
toggle enlargement=evenpage,
size=minimal,
colback=white,
before upper={\parindent=0pt},
before skip balanced=1pt,
after skip balanced=0pt,
extras={frame empty}}
\begin{document}
\section{First}
\textbf{Normal text with standard margins.}
\blindtext[1]
\beginnumbering
\begin{criticalmarginsbox}
\pstart
\textbf{Text with narrower margins:}
\blindtext[1]
\pend
\pstart
\textbf{A second paragraph} in the same block to demonstrate that that the margin size is only page-sensitive.
\pend
\end{criticalmarginsbox}
\endnumbering
\textbf{The normal text has standard margins again.}
\blindtext[1]
\end{document}

\newgeometryfrom the geometry package on the relevant pages, or just in conjunction with\beginnumbering. – marquinho Feb 09 '22 at 12:41\beginnumbering\leftskip=2.5em\rightskip=2.5em\pstartwould help with the main text (not the footnotes). The call of\endnumberingresets the layout, so the next non-critical text has normal margins again. However, this won't work for the footnotes: parameters like \Xwidth[series]{\pstart\noindent(this is a general thing, also outside ofreledmac). (1) is more complicated and my current suggestion cannot handle it. (Briefly put: the margin lengths are set when\beginnumberingis called. If the "critical" block breaks to the next page, it still uses the same lengths – it cannot switch left with right margin.) Could you please edit the question to include this request? This increases the chance that proficient users see it and find a solution :) – marquinho Feb 09 '22 at 14:46