As above, when, sadly, the first parent dies, the remaining parent will be next of kin so, with some exceptions, everything passes to them and it will be them making decisions about the funeral.
Don’t assume because they are older (and mid-70s isn’t particularly elderly) they won’t be able to do this. You don’t mention either parent being unwell or having dementia, and it sounds like they are managing their own lives, plus supporting you and your brother, so it is more likely than not they will be capable of this.
(Even if the remaining parent has a dementia diagnosis by then, they may still be able to contribute as capacity isn’t always completely lost in all areas, especially in the early stages, and they may remember details like what their spouse has said about funerals as it would be in their long-term memory.)
When the time comes to plan the second funeral, go back to the same company and ask for the same thing, with whatever tweaks you want; they will have the record of what was done and you’ll have a good idea of what it will cost. If that parent still had the house when they died, it will need to be sold and, unless it is left to just one of you, that will need to be done jointly with your brother.
There is a Tell Us Once service for informing government departments about a death and banks etc have bereavement departments - you won’t need (and are prohibited in law) to access their accounts with their passwords.
Have you made a will? It’s probably more important that you do this than your parents do, as whoever is left behind will need to make arrangements for the care and financial support of your children, and doing so may encourage your parents to at least have the conversations with you.