The definition with argument can use \ignorespaces at the end of the definition text to prevent that the following space matters in horizontal mode after \noindent:
\documentclass{minimal}
\setlength{\parindent}{0 pt}
\setlength{\parskip}{12 pt}
\newcommand*{\mySpace}[1]{\par \vspace{#1pt} \noindent\ignorespaces}
\begin{document}
Hullo world.
\mySpace{1}
Hullo world.
\renewcommand{\mySpace}{\par \vspace{1pt} \noindent}
Hullo world
\mySpace
Hullo world
\end{document}

Case with argument
The original definition:
\newcommand{\mySpace}[1]{\par \vspace{#1pt} \noindent}
\par ends the paragraph, current mode is vertical, where spaces are ignored.
The space after \par is already ignored as it ends the command token.
There are two kinds of command token, which start with a backslash:
Long form, where the name consists of letters (one or more) (\par, \a, \vspace, ...).
Short form, where the name consists of one non-letter character (\,, \:, ...).
In the first case, a following space is ignored, in the latter case it is not, because TeX already knows, that the name cannot contain more characters.
The space after \vspace{#1pt} becomes a space token, but it is ignored, because TeX is still in vertical mode. The mode is now changed by \noindent. Thus the space after \mySpace{1pt} matters and it can be seen in the output.
This space is ignored, when \noindent is followed by \ignorespaces.
BTW: #1 is expected to be a number, not whole paragraphs, then the argument can be made non-\long, which means that TeX complains during the scanning of the argument, if it find empty lines or \par tokens. The arguments of a command are made non-\long, if the star form of \newcommand is used.
Case without argument
The space after \mySpace is ignored as space after a command name, see above.
\myspace{}and you will see the space appear there as well. – Steven B. Segletes Oct 16 '15 at 23:59%, as suggested in What is the use of percent signs (%) at the end of lines? – Werner Oct 17 '15 at 00:02