You can use:
\setchapterpreamble[o]{\pdfbookmark{Second Chapter}{2nd}}
before \chapter*
to have a bookmark with the anchor just before the chapter heading but already on the new page. So, if you select the bookmark, the PDF viewer will jump to the beginning of the chapter heading:
\documentclass[twoside]{scrreprt}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{blindtext}
\usepackage[]{hyperref}
\begin{document}
\tableofcontents
\chapter{First Chapter}
\Blindtext%
\setchapterpreamble[o]{\pdfbookmark{Second Chapter}{2nd}}
\chapter*{Second Chapter}
\Blindtext%
\Blindtext%
\end{document}
See the KOMA-Script manual for more information about \setchapterpreamble
.
Another suggestion would be to add an explicit \cleardoublepage
before the \pdfbookmark
in you example:
\documentclass[twoside]{scrreprt}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{blindtext}
\usepackage[]{hyperref}
\begin{document}
\tableofcontents
\chapter{First Chapter}
\Blindtext%
\cleardoublepage
\pdfbookmark{Second Chapter}{2nd}
\chapter*{Second Chapter}
\Blindtext%
\Blindtext%
\end{document}
Here the anchor is at the top of the text area, so you have some more white-space above the chapter title, if you jump to the bookmark:

The difference in both suggestions varies depending on the vertical position of the chapter head, e.g., if you would use something like
\RedeclareSectionCommand[beforeskip=-.5\textheight]{chapter}
I would use the first suggestion.