One way to do this is to adjust the fourth argument supplied to \@startsection as part of article's definition of \paragraph:

\documentclass{article}
\setlength{\parindent}{0pt}% Just for this example
\begin{document}
Hello
\paragraph{A paragraph}
Some more text
And we continue.
\bigskip
\hrulefill
\bigskip
\makeatletter
\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}%
{\parskip}%{3.25ex \@plus1ex \@minus.2ex}%
{-1em}%
{\normalfont\normalsize\bfseries}}
\makeatother
Hello
\paragraph{A paragraph}
Some more text
And we continue.
\end{document}
My reference of reducing the space before "very slightly" was to take it down from 3.25ex \@plus 1ex \@minus .2ex to \parskip (0pt \@plus 1pt).
As a reference, see Where can I find help files or documentation for commands like \@startsection for LaTeX?.
\paragraphis the 4th level section heading, it's wrong to use that without\section...\subsubsectionbut assuming this is just an example, copy the definition of\paragraphfromarticle.clsinto your preamble and reduce the lengths – David Carlisle Jul 03 '14 at 20:00\subsectiondefined using\@startsectionin werner's answer but with{2}instead of{4}or if it's a kind of ad hoc heading not really part of the hierarchy you could just use\textbf{}– David Carlisle Jul 03 '14 at 20:46