Yes, you can use the Global pass on Eurostar, but it costs an extra €30 for compulsory seat reservation. These are limited in number, so your parents should buy the reservation as soon as they can.

They will also need to pay €18 if they take the cross-border ICE/ TGV trains from Paris to Germany. (Other routes are possible)

No - city transport (bus/ metro/ tram) is never covered by Eurail. In German and Swiss cities, you can use the S-Bahn (regional/ commuter trains) which have several stops in the city. But they will probably want to buy a city ticket. In London, it’s valid on the Elizabeth Line, Thameslink and Overground, but not the tube.

If you’re living in Germany, you might need to buy the Interrail pass. Similar to Eurail, but it restricts how much travel you can do in your own country (Germany). It depends on how long you’ve been in Europe. Details here:

https://www.eurail.com/en/help/interested-in-eurailing/do-i-need-a-eurail-or-an-interrail-pass

And lots of useful information on Eurail here

https://www.seat61.com/how-to-use-a-eurail-pass.htm

If you plan to travel around Christmas and New Year, please get your seat reservations as soon as possible, especially Eurostar (London - Paris), as there are no good alternatives for that connection.

Your parents should definitely consider buying regular tickets from London to Stuttgart instead of a rail pass.

Ticket 1 London-Paris return, booked on www.eurostar.com (starts at €78 return, even less in a sale)

Ticket 2 Paris-Stuttgart return, booked on www.bahn.com 

This could well be cheaper than using three pass days, plus paying €100 in reservations per person

I'd say it depends on the period + what they want to do in Switzerland.

Around Christmas & New Year Eurostar tickets are well over 100€ one-way.

Are you making a day trip to Switzerland ? Stuttgart - Zurich is already a 3h journey (when the Gäubahn is open) so you won't have much time there.

The ideal option would be to go straight away from Switzerland to Paris and London.

Yes most Swiss S-Bahn services are covered but 
Carefull Swiss S-Bahn operated by SZU mainlyfrom Zürich  up to Uetliberg and Silthal are not covered :) 

Thank you so much. This is very helpful !

Yes I think this would make more sense, As it's only a week we don't plan to do too much or cramp it

Yes, but the only reason I am considering the pass is because of its flexibility to travel anywhere without making a plan in advance, which is what I'm looking for (i.e after they enter the EU). But like you said I'll compare the seat reservations charges plus the pass alongside the individual ticket prices. Thank you ! 

Yes the pass will give you a lot of flexibility in Switzerland and Germany !

But trains with mandatory reservations need to be booked well in advance, especially the Eurostar. This reduces flexibility but it is what it is. When are you looking to travel ?

Check Eurostar availability here : https://www.b-europe.com/EN/Booking/Pass#TravelWish

TGV/ICEs between Paris and Stuttgart are also very popular but you've got alternatives (multiple TGVs to Strasbourg) as opposed to the Eurostar.