13 Days, 3 Cities. How long in each?

Ignoring your arrival day, which will be spent getting over jet lag and your departure day, you actually have 12 days on the ground.

If I was starting with a blank piece of paper, I would divide the time equally between London and Paris. However, adding in Barcelona, I would head to Paris on the 14th and take the Eurostar to London on the 18th or 19th. Moving location loses you at least half a day.

Eurostar tickets are already available, so book as soon as you have confirmed your dates. Prices only head in one direction.

Paris has the best food, London has the most interesting day trips and London is easiest if you don’t speak Spanish or French. London and Paris have the best attractions. Many of London’s museums and galleries are free.

Oh, about that jet lag...forgot to mention we'll be arriving in Barcelona from Tampa, Florida by ship, so no jet lag. That means we'll actually be able to enjoy the day of arrival. That said, your recommendation seems to be the most favored so far, though there's a definite school of thought promoting Catalan food. I didn't get to eat much when I was last in Barcelona but I did in Madrid and I have to say, tapas are wonderful.

I could easily spend a week in each city (and that's after many days in each over many years) and not run out of things to do. I can tell you what I like best, but that is probably not what you'd most enjoy. The best thing for you to do is look at what is available and make a list of the things that most interest you.

For Barcelona, start with this official site. For Paris, the Paris Museum Pass site lists all the sights included in the pass with brief descriptions and links to their sites. Note that a few popular things, like the Eiffel Tower are not included.

If you have 13 days, I would spend 3 nights in Barcelona, 5 in Paris and 5 in London.

Barcelona, don't miss Sagrada Familia Cathedral (be sure to go inside). Also take a tour to Monserrat. Take a walking tour of Barcelona or a day tour that goes to the Gaudi sites.

Paris you have the Eiffel Tower, Notre Dame Cathedral, Louvre Museum, Versailles Palace, Sacre Coeur, Seine River cruise, Les Invalides, the left bank, Arch de Triumph, and more.

London, Westminster Abbey, Parliament, Tower of London, St. Paul's Cathedral, British Museum, Churchill War Rooms, day trip to Windsor Castle and more.
https://travel.usnews.com/London_England/Things_To_Do/

Really depends on your interests. One could easily spend a full week in each city, there isn’t one that screams “short visit!!”

For me:
Barcelona the most. I personally love the modernista architecture, the creative food, and the energy of this city.

Then London - it is huge, so much to see. I could spend a full day at the British Museum. Pubs are fun and there’s good food (not a traditionally British thing, but a modern city thing!). If you like theater, go in London!

Then Paris. Yes, it’s Paris. It’s beautiful. It’s famous. The food is amazing (but not my personal favorite type). The museums are world class. But there’s a little bit of stiffness and formality that wears on me. This isn’t a “French people are rude” thing - they are not, everyone is very polite. It’s just an atmosphere thing. I like Paris, I just have more fun in Barcelona and London.

I agree with the PP who said 3 Barcelona and 5 each in London/Paris. However, all 3 of those are big cities that are spread out and you will likely get some touring fatigue. Have you considered doing just 2 big cities and then having your 3rd city, between them, be somewhere relaxing, like the beach, or Swiss Alps, or Provence, etc.?

We actually did consider just London and Paris, but since the ship docks in Barcelona, it seemed a shame not to have a decent look around. I do agree, though, that London and Paris should get the lion's share of our time in Europe. God, I miss having all the time in the world! I was there for four years, saw virtually every country in Europe (except Finland) and easily could've spent another 20 years over there without being bored or lonely.