Speed train

Avoid Rail Europe at all cost, buy the tickets from direct from the company that operates the trains. In your case that is SNCF, www.sncf-connect.com and in case you have problems with their website there are better 3rd parties than Rail Europe. Just search for a ticket from Nice to Bayeux and you will see the options.

The general rule for luggage on trains is that you can bring as much as you can carry yourself. No difference between 1st and 2nd class. Although for the low cost trains, Ouigo, I think there are some additional fees.

I'd suggest getting large backpacks in place of the carry ons which will make managing large pieces of luggage on trains easier. Take a TGV to Paris and then whatever to Bayeux. You can carry the suitcases on the train but stowing them may be tricky as there is limited luggage storage space. Be on the train as soon as boarding opens (usually 15-30 minutes before it leaves the station and presumably Nice is the starting point so you can get your stuff stowed early.

From Paris to Bayeux you will need to move from Gare de Lyon to St. Lazare -- probably with big luggage you will want to do that by cab.

Well I guess I would hate lugging 2 big parachute bags all over France if I didn't need them. You might find a place to store them in Paris.

There are direct TGV's from Nice to Paris. In Paris you will have to retrieve the parachutes and go to Gare St Lazare for trains to Bayeux.

Thank you so much for your suggestions.

Have you considered just flying from Nice to Caen? There's a direct flight on the 31st that will get you there in less than 2 hours versus the long, long day via the train - not to mention saving you the hassle of horsing parachute bags all over the country.
Air France has a few flights on the 30th ... connecting thru Lyon.
I don't know how prices compare versus train travel, but the convenience alone might make it well worth your while.

We looked at plane, train and automobile. Let just say that the parachutes are a nightmare but this is a once in a lifetime event for the hubby. We just booked the train so we can see the country.

I have to admit that I have not idea how big or heavy a parachute is, but do you need to carry them with you for the rest of the trip?

I just found this thread too late. For that long long train trip diagonally across the full length of France I would have taken the tram just down the Promenade des Anglais to the Nice airport, easy quick flight to Caen, then simple short quick train to Bayeux - or rent a car - rather than the somewhat complicated slow train - it isn't quick until it gets away from the Riviera - to Paris gare du Nord, then across town to gare St Lazare, then the train all the way to Bayeux.

I'm sorry I couldn't catch you in time...

You will arrive at Gare de Lyon, then have to go to Gare St Lazare. If there is a Metro line that i would say you could handle with a lot of luggage, the No 14 line would be the one. Since it is deep underground, all the stations have elevators, and it is somewhat of an express line. There are only 3 stops between GDL and GSL.

I still think storing the bags when you arrive in Paris and picking them up on the way to Bayeux. Nannybag has storage service, insured, for 6 EUR per day, so 2 weeks (15 days) is 90 EUR per bag. https://www.nannybag.com/en/

With all that luggage, forget the idea of taking public transportation between Paris train station. (Seriouisly, sometimes people need to get out of the "cheapest" possible mindset.)

Take a taxi. You will find them at the taxi rank. Do not go with anyone who comes up to you in the station. Go to the official taxi line. There will be signs.

With all that luggage on the Metro, you will get the attention of the thieves while you struggle. Take a taxi.

Thank you all so much for the wonderful suggestions and tips. We will definitely look into using the nanny bag.

Nigel's Nice to Caen flight suggestion is excellent.