What would be a suspicious sum of money to pay into another person's UK bank account ?

There's usually no issue if using bank transfer, up to 10K or depositing a cheque.

Paying in cash is no longer possible unless you are the account holder (& have ID) 

The government would love to get rid of cash because they want to keep an eye on what you and I do with our money (and how much we have.  They think it's their business!).

Also once 'your' money is just a number on someone else's computer it is very easy to flick a switch and stop you from accessing it.

Electronic money will stop some frauds but will make other types of even bigger theft more possible.

Someone once said "If you can't put it in your pocket   you don't really have it."

It's not the amount that suspicious, it's the source. I've had over £80k paid into my account this year in three sums and not one was questioned as the sources were legit.... But try paying too much cash into someone's account and you come unstuck.

Dad needs to speak to his bank. If it's legit money, then he has nothing to worry about

Then your dad couldn't prove where the cash was from, where as a bank transfer of money proves it was already in an account and so legal/traceable. ALL and anything over £10k is automatically reported to HMRC by all and any bank

Then you and Dad  with his ID should have gone to your bank and explained.

They are open.

U.K.

Usually if you keep it under 10K

You need to talk to your bank, to explain. Banks are instructed to spot/report suspicious money laundering  transactions. If your account normally received large deposits, it would not be questioned.

Banks have to ask questions to prevent money laundering. I expect your dad's bank asked for the source and he declined to answer. When I paid off lump sums on my mortgage I also had to describe the source. So that's what needs sorting out.

Anything more than £8,800 can be randomly investigated, they will still eventually make the transfer. You can't get around it making a series of smaller transfers, they look for that too. It may change soon because it was EU policy.