We put 'byo alcohol' on the invite, and the time was '4pm till late' and it was a BBQ dinner.
Jasper's 3rd b'day is in a fortnight and the friends I've invited will know its byo if they want to drink.
And yeah I wouldn't have alcohol at a 'kids' party where parents are leaving their kids.
I guess if anyone is ever offended that we drink a couple of wines with a BBQ dinner for jasper's birthday and I haven't specified this on the invite they can leave, I would maybe be surprised, but as long as they didn't think badly of me I wouldn't be offended.
As for other children's parties, it would really depend on who's party it was. If it was in the early evening and I wasn't sure and didn't want to put them on the spot by asking (in case they felt uncomfortable or thought I was being inappropriate asking!) I'd probably bring wine in the car or stashed in a bag and not bring it out unless others were drinking.