One Day in London : Need Help!

The order in which you've listed the sights is kind of circular; would it be less walking and a better use of your time to make it more linear? I'm not sure. Another factor is the timing of your brunch and Buckingham Palace arrival. You might be getting there as the changing of the guard is happening, in which case it will be mobbed and you likely will see very little. When we were there we followed the RS book's advice and watched the other end of the guard change, at St. James's Palace. A crowd gathers there, too, but it's a much smaller one.

Of course everything depends on what your parents particularly want to see and do in London. If this list of sights to see comprises their targets, then I'd say stick with it as much as you can. Be flexible in case your morning flight is late, or it takes longer than you expected to get into town from the airport.

When you say "see" Buckingham [Palace]: you will not have time to take the tour inside and still visit all the other destinations on your one-day list, unless you just go whizzing past them. One could easily spend all day there. https://www.travelawaits.com/2471433/how-to-visit-buckingham-palace/

Westminster Abbey is right by the Houses of Parliament and Parliament Square, the latter featuring some memorable statuary and a view of the [Coca-Cola] London Eye across the Thames. For us the trick was figuring out that when you exit the Tube at Westminster station, you pretty much have to walk all the way around the building to find the entrance.

Hope this helps!

Thameslink is cheaper but the Gatwick Express is faster.

Personally with such a short amount of time I’d look online how to get Gatwick Express tickets bargains.
Like the Heathrow Express you can save lots of money by booking at least 30 days out. Research.

Then take the Express straight to Victoria Station. There’s a Giraffe restaurant nearby, perfect for a full breakfast. If you just need coffee and a pastry try TomTom coffee. Back at Victoria take a District or Circle Line Underground Train to Westminster.

Exit into Parliament Square. Even with Elizabeth’s Tower in scaffolding and Big Ben silenced you will definitely know you are London.

Spend time in Westminster Abbey ( take the Verger’s tour) . Depending on your timing maybe lunch in the Abbey Cafe.

Then walk up Birdcage Walk through St James Park to see Buckingham Balance. From there walk through Green Park. Do they want to shop or have tea. Fortum and Mason ticks both boxes.

I’m thinking this is mid day not yet dinner time. The hustle and bustle Picadilly as you approach Picadilly Circus then through Leicester Square passing the National Gallery back into Trafalgar Square will definitely permeate everyones senses.

Now time for dinner and because I’m a firm believer first time visitors should enjoy pub grub make your way to the intriguing Ship and Shovel Pub.

Then take a District or Circle underground train from Embankment to Victoria and then back to Gatwick on the Express.

Perfect “brief” visit.

Edited: and you are welcome

Why not fly into London City Airport? There are a number of direct flights from Dublin. You could then start at the Tower and work out from there to Parliament Square and Buckingham Palace.