I've organised several alternate shot Father-Son/Parent-Child events. They worked okay with different handicap divisions playing from different tees, although players need to be aware that standing in an inappropriate place could result in you getting a ProV1 in the back of the head! (And I'm not being just jokey when I say this!)
We found a pairing of a long hitting but higher hcp kid and a low hcp savvy golfer parent normally wins, which can make it a bit of a chore for those with no chance of getting a prize, so I'd suggest having a range of other prizes such as straightest tee shot (paint a line in a fairway), nearest the pins, any holed bunker shots etc. Still Father-Son and Parent-Child should mainly be for fun IMO.
An alternative is what is known in some places as a Greensome - both players tee off, the team then decides which ball to play for the 2nd etc shot and you play alternatively from there. A bit safer but once again, we found a pairing of a long hitting but higher hcp kid and a low hcp savvy golfer parent normally wins.
Some adults tend to get rather over-competitive and do things like change ball type depending on who is playing the shot. Watch out for this kind of behaviour and stop it, otherwise things can potentially get unpleasant!
I recommend no gross prizes though or maybe just one and then just a minor one.
Grandparent-Grandchild pairs is also a terrific format, again one which I've organised several times, and thoroughly recommend.
For much younger kids/juniors many of whom arn't yet able to walk/behave for 18-holes consider holding the same event over say 6-holes or 9-holes. Also hold fun events around the Clubhouse like chipping through hoops or into inflatable paddling pools or putting into painted circles of different diameter as these tend to keep little ones active so they don't get bored, misbehave and/or pester the adults, coz if they do the 'powers-that-be' around the club tend to not want you to hold the event again.
I'd also suggest you combine all this with a BBQ and have a big prize giving ceremony at the end with the Club Captain giving out the prizes. Take lots of photos too and post them on your clubs website/noticeboard. Parents/grandparents love seeing their kids receive even a minor prize and then they want to participate again next year. And as to the famous 'growing-the-game' phrase, well nothing keeps youngsters wanting to carry on playing than getting a prize that they can keep on a shelf in their bedroom and look at regularly.
Have fun, for as I said above, that's what Father-Son and Parent-Child pairs golf should mainly be about.
atb