For seat selection, that should be triggered once Assistance is requested and once the ticket is issued. Travel agents tend to ticket at the last moment for cashflow reasons and that may explain why you can do it at the moment. I would suggest the traveller concerned should ring the BA Assistance team in Newcastle, open office hours but also Saturdays, on this number:
0800 408 1100
to check both the assistance and seating issues.

If this is for the summer of 2025 I think I can see the problem. There are 3 flights a day most days between London and Miami which have BA flight numbers on them. But only the first service, BA207 deparating LHR at 09:55, is BA operated and from Terminal 5. The two later services around midday and 14:00 hrs, are codeshares with AA out of T3, which firstly makes the journey a lot more difficult if assistance is involved and secondly makes it more awkward for issues such as seating, ticketing, check-in. It may be that the travel agent can still change the flights, to make it a much easier connection, but that comes at downsides to your parents and possible extra costs. One downside is that your parents may need to leave EDI on a flight at 06:00 or 06:30 to connect on to BA207. This will keep the connection within T5. Knowing how travel agents work, they may prefer your parents to leave the night before, and let the TA book a hotel in Heathrow, which would be another option.

For upgrades, the most cost effective way is to check at the airport on the day of departure as to whether BA or AA is selling upgrades to the next cabin. If it's a Trade booking, in theory you can do upgrades using Avios but it may be relatively expensive and not easy to do. If it's a cruise related package it's very likely that the most cost effective solution is to pay for the new cabin at the point of booking - by the time you add up the total cost of airport upgrades it's often more expensive than the cheapest premium cabin ticket at the time of booking. It just looks affordable by being split into smaller chunks.

By all means check back here before accepting your TA's suggestions.