One possibility is that you might need write access to the registry key HKEY_CLASES_ROOT.
Assuming perl is installed in C:\Perl, from the cmd.exe command line:

SET PATH=%PATH%;"C:\Perl"
ASSOC .pl=Perl
FTYPE Perl="C:\Perl\bin\perl.exe" "%1" %*
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If you find this is only temporary, when it does not work then use the ASSOC and FTYPE to see what else is happening, if anything. Failing that you can always put the above into a .bat file and run it from a short-cut with something like:

%comspec% /k "C:\runperl.bat"
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that's what I do when I have more than one Perl version installed.