It's not the accepting cash from your parents that is abnormal.
It's not the dad wanting to give cash to his daughter that is abnormal.
It's that somehow the dad and the mum have a MARITAL issue about it - for whatever reason - and the OP is unwittingly complicit in it.
If OP's DM doesn't want her DH to keep giving handouts to their DD, that is between them. But somehow, OP knows that her DM is not speaking to her DF because of it.
I strongly suspect this is because the DF is speaking out of turn and therefore is behaving disrespectfully to his wife, regardless of whether it is "normal" to help out offspring on £40K a year.
What should be going on is:
DF gives money to DD.
DM says "Oi, you should stop that for X reason"
[where X could be anything at all - not being fair to other DC by doing the same; leaving them as a retired couple short for something they meant to buy/do; on principle that DD has had X amount recently already]
DF apologises to his wife for doing it without consulting her.
The end.
What seems to have happened is that DF has either ignored his DW in the past on this, or wants to be the 'good guy' so thinks he's free to bring his DD into disputes with his DW.
DD is an adult, as are her parents. She shouldn't be privy to their disagreements about money etc. Why is she? Because her DF told her. Thus putting her DM in a worse position than if she's never mentioned it.
DM becomes the "bad guy" who is "tight".
It's not really about whether adult children get money still from their parents.
It's about respect, as far as I can see.