Here's my version:
\documentclass{article}
\usepackage{mdframed,lipsum}
\parindent=0pt
\newmdenv[leftmargin=\dimexpr-0.5em-3pt, innerleftmargin=0.5em,
rightmargin=\dimexpr-0.5em-3pt, innerrightmargin=0.5em,
linewidth=3pt,linecolor=red, topline=false, bottomline=false,
innertopmargin=0pt,innerbottommargin=0pt,skipbelow=0pt,skipabove=0pt,
]{notex}
\newenvironment{note}
{\par\vskip\dimexpr\dp\strutbox-\prevdepth\relax\notex\strut\ignorespaces}
{\par\xdef\notetpd{\the\prevdepth}\endnotex\vskip-\notetpd\relax}
\begin{document}
\leavevmode\llap{\smash{\vrule depth7\baselineskip height0pt\hskip1em}}%
\lipsum*[2]
\begin{note}
\lipsum[2]
\end{note}
\leavevmode\llap{\smash{\vrule height\baselineskip\hskip1em}}%
\lipsum[2]
\end{document}
The black rules are just to show that the alignment is correct (they can't be too near the text because mdframed applies a white background).
A magnified view of the top line
And one of the bottom line
At the start we ensure to skip by the glue TeX would insert and we add a strut to the first line in the note environment, so to have the correct distance between baselines. At the end we do similarly, keeping into account the depth of the last line in the note environment.
Brief explanation
mdframed sets its chunks in a \vbox which is inserted in the main vertical list; this usually upsets the interline spacing, as a \vbox has its reference point at the baseline of the last box inside it. The package tries to do clever things not to add unwanted spaces, but in this case it's not sufficient.
So we need to emulate TeX's normal behavior. The distance from one baseline and the next is the sum of \dp\strutbox and \ht\strutbox; we can insert a strut in the first line of note which makes it the correct height; but adding \dp\strutbox vertical space will usually be too much, because the last line before note may have descenders. So we end the paragraph so that \prevdepth (an internal parameter) contains the depth of the last contributed box (precisely the last line). So we can add
\vskip\dp\strutbox-\prevdepth
and all is good. Something similar we have to do at the end. We end the paragraph and define \notetpd to contain the value of \prevdepth, so we can remove a vertical space with that amount, putting everything back in synch.
Best Answer
\paragraphis (despite the poor name) not related to paragraphs but rather is a section heading, following subsection and subsubsection in the hierarchy. It shares all the heading code for automatic numbering and adding entries to the table of contents, although numbering is by default not used at this heading level in the standard classes.So if this paragraph is not in a new sub-section and you just want a font change then simply
Is perhaps the most natural markup.
While it would be possible to redefine
\paragraphit is usually strongly advised not to redefine or change a journal class layout code as it can complicate (or prevent) inclusion of your document in the journal or proceedings if your definitions conflict with the overall processing of the combined work.