Below are two possible solutions. I wrote the bottom one first, and it is a simple extension of your original code that does the same thing for \itemize that you had already done for \section. I think the top one is probably more durable in the sense that it seems less likely to break (and easier to adapt), but things might've been set up as they were for some other reason that I'm not aware of.
New solution:
Here is a new solution, which I like better:
\documentclass[11pt]{article}
\usepackage{titlesec}
\setcounter{secnumdepth}{3}
\usepackage{etoolbox} %% for \apptocmd and \pretocmd
\setlength{\parindent}{0pt} % do not indent subsequent paragraphs
\newcounter{para}
\newcommand*{\numberedparagraph}{%
\refstepcounter{para}%
\makebox[0pt][r]{\textbf{\thepara.~}}% %% This puts the number in the margin
%\makebox[0pt][r]{\makebox[0pt][l]{\textbf{\thepara.}}\hspace{2em}}% %% This also left-aligns multi-digit numbers
%\textbf{\thepara.~}% %% And this indents the paragraph instead
}
\let\oldep\everypar
\newtoks\everypar
\oldep{\the\everypar\ifinhibitparnums\numberedparagraph\fi}
\newif\ifinhibitparnums
\inhibitparnumstrue
%% Inhibiting paragraph numbers in section/subsection/... titles
\makeatletter
% \pretocmd{\@sect}{\inhibitparnumsfalse}{}{} %% <- this only works without titlesec
% \apptocmd{\@sect}{\inhibitparnumstrue}{}{} %% <- this only works without titlesec
\pretocmd{\ttl@straight@ii}{\inhibitparnumsfalse}{}{} %% <- this only works with titlesec
\apptocmd{\ttl@straight@ii}{\inhibitparnumstrue}{}{} %% <- this only works with titlesec
\makeatother
%% Inhibiting it in itemize
\pretocmd{\itemize}{\inhibitparnumsfalse}{}{}
%% N.B. \ifinhibitparnums is reset automatically when the environment ends
\begin{document}
\section{Section 1}
This is paragraph 1.
This is paragraph 2, and here are some bullets:
\begin{itemize}
\item One.
\item Two.
\item Three.
\end{itemize}
And then this paragraph should be numbered 3, and it is! Also it is long enough to span two lines!
Numbering returns for this paragraph, and the count is still correct.
\end{document}
I defined a conditional, \ifinhibitparnums and number paragraphs only when it is set to false. I prevent section titles from being numbered by redefining \ttl@straight@ii, which produces section titles for titlesec, so that it first sets this conditional to true, then does its thing and finally sets \ifinhibitparnums back to false. (To be more precise, I use \pretocmd to prepend \inhibitparnumstrue to this macro and \apptocmd to append \inhibitparnumsfalse. These are from the etoolbox package.)
You can also try to for instance similarly patch the tabular environment using \pretocmd{\tabular}{\inhibitparnumstrue}{}{}. This can also be made to work for starred environments, as explained in this answer.
To temporarily inhibit paragraph numbers, you can also manually type \inhibitparnumstrue, and to resume numbering type \inhibitparnumsfalse.
Here is the output:
Original solution:
The value of \everypar is temporarily changed after the end of an environment, just like after \section. This is done by the internal macro \@doendpe, which is called by \end.
One possible solution would thus be to redefine \@doendpe in exactly the way as you had already modified \@afterheading:
\documentclass[11pt]{article}
\usepackage{titlesec}
\setcounter{secnumdepth}{3}
\setlength{\parindent}{0pt} % indent subsequent paragraphs
\newcounter{para}
\newcommand*{\numberedparagraph}{%
\refstepcounter{para}%
\makebox[0pt][r]{\textbf{\thepara.~}}% %% This puts the number in the margin
%\makebox[0pt][r]{\makebox[0pt][l]{\textbf{\thepara.}}\hspace{2em}}% %% This also left-aligns multi-digit numbers
%\textbf{\thepara.~}% %% And this indents the paragraph instead
}
\let\oldep\everypar\newtoks\everypar
\oldep{\the\everypar\everypar{\numberedparagraph}}
\makeatletter
\renewcommand{\@afterheading}{%
\@nobreaktrue
\everypar{%
\if@nobreak
\@nobreakfalse
\clubpenalty\@M
\if@afterindent
\else
{\setbox\z@\lastbox}%
\fi
\else
\clubpenalty\@clubpenalty
\everypar{\numberedparagraph}% <- modification
\fi
\numberedparagraph% <- modification
}%
}
\def\@doendpe{\@endpetrue
\def\par{%
\@restorepar\everypar{\numberedparagraph}% <- modification
\par\@endpefalse
}%
\everypar{%
{\setbox\z@\lastbox}\everypar{\numberedparagraph}% <- modification
\numberedparagraph % <- modification
\@endpefalse
}%
}
\makeatother % <- you forgot this, it is good form to restore the catcode of @
\AtBeginDocument{\everypar{\numberedparagraph}}
\begin{document}
\section{Section 1}
This is paragraph 1.
This is paragraph 2, and here are some bullets:
\begin{itemize}
\item One.
\item Two.
\item Three.
\end{itemize}
And then this paragraph should be numbered 3 but it is not.
Numbering returns for this paragraph, but now the count is incorrect.
\end{document}
It feels like there should be a better solution though (edit: see above). (One that doesn't break whenever LaTeX does anything with \everypar.)
