Almost always an activator posts the park they'll be operating from on the POTA website (date and approximate start time).  Then, when the activator begins calling CQ POTA, nearly 100% of the time the hunters know which park the activator is operating from.  From that point, typically the hunter will spot the call and frequency that the activator is operating on using the POTA website.  POTA also includes the park number (US-3615 for the Dade Battlefield, for example).  Spots also come from automatic spotters which are populated on the POTA website (https://pota.app).  I know I've seen spots of me from a number of international spotting "machines" which include my signal strength in dBs.

So, for most activations, the activator does not have to identify the park.

BUT, assuming (as you have pointed out), that you hear an activator calling CQ POTA without identifying the park, just ask the op what park they're operating from and they'll ID it.  From personal experience, I've been asked for my park number and I immediately tell the hunter what my POTA park ID is.

Expanding further:  Let's say to have a QSO with an activator, then within 24 hours (and often times sooner), the activator uploads their log to the POTA website.  Then the POTA website links to the hunter's account and gives you credit for that contact which includes full details of the QSO.  In a way the POTA website acts as a logging, spotting and LoTW program wrapped in one bundle (once the hunter creates an account within the POTA website (it's free).

If you haven't checked out the POTA website, please do so.  There are many "rewards" for making QSOs both for the hunter as well as the activators.  It's very similar to working all counties, states or countries.

If you have further clarifying questions, just ask me.

Too bad you can't join me on Friday.  When I return to TV in November, I'll post an announcement on the TVARC website that I'll offer another local POTA activation for TVARC members (or anyone else).

73, Tom, K3WT