You can use a makeup to set up a custom page layout. Here I used the pre-defined standard makeup, which centres vertically.
\setuphead
[part]
[placehead=yes,
alternative=middle,
before=\startstandardmakeup,
after=\stopstandardmakeup,
header=empty]
\starttext
\startpart [title=Part One]
\stoppart
\stoptext
You can also create your own makeup and use the align key for the vertical alignment. This approach is more flexible. By default makeups do not increment the page counter (thanks to Aditya for pointing that out). This can be fixed with the pagestate key.
\definemakeup
[part]
[align=middle,
pagestate=start]
\setuphead
[part]
[placehead=yes,
alternative=middle,
before=\startpartmakeup,
after=\stoppartmakeup,
header=empty]
\starttext
\startpart [title=Part One]
\stoppart
\stoptext