Update
KOMA-Script version 3.26 introduces two new keys to \RedeclareSectionCommand and \RedeclareSectionCommands: runin and afterindent. Possible values for both keys are bysign, true and false. For more information see the KOMA-Script documentation or eg Adjusting spacing around section/subsection titles with koma-script.
Using the new key runin with value false¹ it is possible to remove the \parskip between the heading and the following text by afterskip=-\parskip.
Example:
\documentclass[12pt,english,parskip=half*,listof=nochaptergap,final]{scrreprt}
\usepackage[utf8]{inputenc}
\usepackage[onehalfspacing]{setspace}
\RedeclareSectionCommands[
beforeskip=-3.25ex plus -1ex minus -0.2ex,
runin=false,
afterskip=-\parskip
]{paragraph,subparagraph}
\usepackage{blindtext}
\begin{document}
\paragraph{Test paragraph}
\Blindtext[2]
\end{document}

¹ Default setting is runin=bysign which results in the same behavior as in the original answer.
Original answer
The vertical space between a heading and the following text is at least the same as the space between two paragraphs in the text body. That means if you are using a parskip instead a parindent (because of parskip=half*) the space between a heading and the following text is at least the same as this parskip.

Code:
\documentclass[12pt,english,parskip=half*,listof=nochaptergap,final]{scrreprt}
\usepackage[utf8]{inputenc}
\usepackage[onehalfspacing]{setspace}
\RedeclareSectionCommands[
beforeskip=-3.25ex plus -1ex minus -0.2ex,
afterskip=1sp,% smallest possible positive value
]{paragraph,subparagraph}
\usepackage{blindtext}
\begin{document}
\paragraph{Test paragraph}
\Blindtext[2]
\end{document}
A positive value of afterskip for a section command enlarges the vertical skip by this value. With negative value of afterskip the text the starts in the same line as the heading and the value of afterskip is used as a horizontal skip.

Code:
\documentclass[12pt,english,parskip=half*,listof=nochaptergap,final]{scrreprt}
\usepackage[utf8]{inputenc}
\usepackage[onehalfspacing]{setspace}
\RedeclareSectionCommands[
beforeskip=-3.25ex plus -1ex minus -0.2ex,
afterskip=-1em,% works as horizontal skip of 1em
]{paragraph,subparagraph}
\usepackage{blindtext}
\begin{document}
\paragraph{Test paragraph}
\Blindtext[2]
\end{document}
So remove parskip=half* from your class options. Then parindent is used instead parskip and only the positive value of afterskip is used as vertical skip between the heading and the following text.

Code:
\documentclass[12pt,english,
%parskip=half*,% <- commented, so parskip=false is used
listof=nochaptergap,final
]{scrreprt}
\usepackage[utf8]{inputenc}
\usepackage[onehalfspacing]{setspace}
\RedeclareSectionCommands[
beforeskip=-3.25ex plus -1ex minus -0.2ex,
afterskip=1sp,
]{paragraph,subparagraph}
\usepackage{blindtext}
\begin{document}
\paragraph{Test paragraph}
\Blindtext[2]
\end{document}
If you really want or need parkip=half and remove the parskip between the paragraph heading you could use
\documentclass[12pt,english,parskip=half,listof=nochaptergap,final]{scrreprt}
\usepackage[utf8]{inputenc}
\RedeclareSectionCommands[
beforeskip=-3.25ex plus -1ex minus -0.2ex,
afterskip=1sp,
%indent=0pt
]{paragraph,subparagraph}
\usepackage{xpatch}
\xapptocmd{\sectionlinesformat}{%
\ifstr{#1}{paragraph}{\vspace*{-\parskip}}{}%
\ifstr{#1}{subparagraph}{\vspace*{-\parskip}}{}%
}{}{}
\usepackage{blindtext}
\begin{document}
\paragraph{Test paragraph}
\Blindtext[2]
\end{document}

But I do not recommend this, because it is unusual to have a smaller space between a heading and the following text as between two paragraphs in the text body.
afterskip=1spyou get the space of the parskip between your paragraph heading and the following text. If you enlargeafterskipthis space is enlarged by the same value. – esdd Feb 10 '16 at 08:46parskipoption is not used while you are settingparskip=half*;-) – esdd Feb 10 '16 at 09:49afterskipchanges the vertical skip to a horizontal skip. So the text would start in the same line as the heading. – esdd Feb 10 '16 at 09:51