Can I run grep against a zOS dataset via Unix system services?

No need to cp first. You can cat, tail, grep etc. The thing to remember is the special quoting and that the DSN is preceded by two slashes:

cat "//'s052WX.EO.REXX(MYREXXSOURCE)'" | grep parse

If you look in the UNIX command reference manual under the cp command there’s more info.

How big a data set are we talking about?
The Unix system services cp command can take data sets as a source, and a Unix file (or directory) as a target, so conceivably if the data set is small enough you can copy it into files and grep away to your heart's content.

As an alternative, if you can open your data set in ISPF on the z/OS machine you may want to consider using FINDRX (which gives ISPF basic regex capability) -- I've never used it myself though so I can't vouch for it working or not.