You can certainly combine Paris and Barcelona in the time you have, but you could also spend all your time in Catalunya. Not only does Barcelona have a wide variety of sights itself, but there are many wonderful side-trips to be made (Girona, Montserrat, Tarragona, Dali sites in Figueres and near Cadaques, Sitges, etc.). So my urgent recommendation is that you get to a book store or library right away and look for a guidebook with good coverage of Barcelona and Catalunya. Don't worry about anything except sights at this point. Does it look like you could happily fill all your time just in that area? Last year, I spent 13 days in Barcelona and Girona with a couple of side-trips, plus extra time near Andorra, but that was because of my particular interests. Your family's interests may differ, but I left Barcelona wanting more.
Next, I'd check first on what flight arrangements might be possible. I like Google Flights, but you may prefer a different website. Choose "multi-city" and see what comes up for flights into Paris and out of Barcelona. Also check the reverse. Then look at round-trips into each of the two cities to see whether there's a substantial extra cost for the multi-city option. I returned home from Barcelona last year without difficulty, but I'm an east-coaster and was using airline miles. Fares are wacko this year, and there's no telling what you'll find. If your dates are somewhat flexible, be sure to check all the possibilities.
Also look at the TGV fares for the potential rail connection. Here you're a bit late in that those tickets have been on sale for at least a little while, and a lot of the bargains may have already been snapped up. You don't want to buy the tickets before you're solid on your travel plans, because the best prices are for non-refundable/non-changeable tickets. But the longer you wait, the more you are likely to pay. The price difference can be really substantial. This is something negative that has occurred since you were last in Europe.
You might also look at Paris-Barcelona (and vice versa) flights on budget airlines. Skyscanner is a good source for that information. Again, tickets for May have been on sale for quite some time, so I don't know how many screaming bargains will be available at this point, but you might be lucky. Be aware, though, that the budget airlines charge for everything and tend to have very low weight and size limits for carry-on bags, so extra fees can really add up. It's important to go all the way through the booking process (short of providing a credit card number), to see what the total cost of a potential flight would actually be.
Once you have an idea of some of the key sights you might want to see in Barcelona (and Paris if you include it), come back here with your list. We can tell you which sights probably need pre-purchased tickets to avoid standing in very long lines. (Another not-so-nice change since you've been in Europe.) I don't think there's anything in Barcelona that has to be booked before you leave the US unless there's a high-traffic sight you plan on Day 1 (not a good idea due to jetlag) or perhaps Day 2, but there may be things in Paris you'd need to work on before leaving home.
