I assume these are very lightweight scooters that can be folded to the size of a carry on. You could start with this link:

https://www.seat61.com/european-train-travel-with-special-needs.htm

Then you need to contact the relevant operator to book spaces.

I have enough experience of Barcelona to wish you the best of luck - it is busy in all pedestrian areas and many of them exhibit a certain intent of purpose. I guess that is not unknown to anybody in a wheelchair or mobility scooter. :

https://www.barcelona.de/en/barcelona-disabled-persons.html

We did though find there are lifts at most, if not all metro stations, but at busy times, like all metros, finding a space to board may be difficult.

I suggest that you contact SNCF and ask for information about what possibilities there are to bring the scooters on the train. Normally there would be limits for the dimensions and weight of the scooter.

https://www.sncf.com/en/passenger-offer/travel-for-everyone/accessibility

As far as I know the metro in Paris is not very accessible. You can find more information about accessibility in Paris local transport here. I can't see any information about scooters.

https://www.ratp.fr/en/accessibility/network-accessibility

You and your wife might be entitled to free Eurail passes as you need to assist your parents. Please read more here:

https://www.eurail.com/en/help/interested-in-eurailing/traveling-with-a-caregiver

Indeed mosttimes the Weight is the problem with such Scooters Most TGV´s have a own Lift onboard Just make sure you are early enough at the station.
 note when you are just doing this trip by train you are cheaper when you book via the Website of SNCF “Normal tickets” especially advance fares can be very cheap :) 

Thank you for the replies. The situation has changed in that my parents have decided not to join us on this trip after all. They decided to keep the originally planned cruise to Hawaii in July which will be over their 69th anniversary instead. My wife and I are still planning on making this trip but obviously do not have the same restrictions. Once again thank you for your input and sharing the links.