A very simple method.
We define a new command
\newcommand{\mathleft}{\@fleqntrue\@mathmargin\parindent}
to simulate the option fleqn with \parindent indenting, and a new command to restore the normal behavior
\newcommand{\mathcenter}{\@fleqnfalse}
So, add the following lines in the preamble
\makeatletter
\newcommand{\mathleft}{\@fleqntrue\@mathmargin\parindent}
\newcommand{\mathcenter}{\@fleqnfalse}
\makeatother
and issue \mathleft before and \mathcenter after the math stuff you want left aligned with \parindent.
You can use this solution with all math environments without redefining any of them.
MWE
\documentclass{article}
\usepackage{showframe}
\usepackage{amsmath}
\makeatletter
\newcommand{\mathleft}{\@fleqntrue\@mathmargin\parindent}
\newcommand{\mathcenter}{\@fleqnfalse}
\makeatother
\begin{document}
\mathleft
\begin{equation}
1+1=2
\end{equation}
\mathcenter
Indented paragraph
\begin{equation}
1+1=2
\end{equation}
Indented paragraph
\end{document}
Output

fleqn? – Ch'en Meng Dec 07 '13 at 13:31$ ... $with\displaystyle(if really need)? – Ch'en Meng Dec 07 '13 at 13:45alignedenvironment, provided byamsmath, works well in inline-math. And its syntax is familiar withalignandsplitwho are used to typeset multiline equations. – Ch'en Meng Dec 07 '13 at 13:51alignedseems to be almost it. Is it possible to control the indent amount of analignedenv and set it to\parindent? – BeeGirl Dec 07 '13 at 13:56alignedis often used as a part of a certain equaion (or maybe the whole part), say block-env. Hence, the answer is no. :( – Ch'en Meng Dec 07 '13 at 14:05flalign*doesn't* mean “left aligned equations”, but “full length align”. Having some display centered and others left aligned doesn't make sense. – egreg Jun 25 '14 at 09:42