TL,DR: your DP is being an arse and you need to stand with your DS on this one (and maybe reset some of the house rules/expectations too).
"...however my partner has flipped out about it saying he's been waiting for my son to offer to pay for his damaged bike or to do some jobs to pay off for the damage. He's so cross that he's getting to 'buy' and pc but not payed for the damaged bike."
What does your partner think this damage amounts to? It's a scratch.
I just ran this past my cycling-enthusiast DH (several bikes, some built from scratch), who said that getting some touch-up paint from Halfords would be a bit shit-looking, scratches you can't be too bothered about, a steel frame might be open to rust if the scratch was very deep but on the whole unlikely, carbon fibre frame a scratch wouldn't matter. Personally he wouldn't bother, but he does know some people who are very precious about their bike who would go full-on respray (hundreds of £).
So, is your partner 'precious' about his bike? Is he a poseur rather than a keen cyclist? Because this 'damage' won't interfere with him using the bike. It will ride in exactly the same way it always has. A touch-up pen was £8 last time I bought one for my car, is that the level he expects - or several hundreds for a respray?
By raising the matter only NOW - he's having a tantrum and needs to grow the fuck up. If he wanted damage paid for, he should have raised it at the time and said how much. An £8 paint pen I might have got DS to pay for, a respray - dream on!
"I've suggested my son doesn't get the pc until he's paid or worked off the bike damage but my parter doesnt want to look like the bad guy."
Bollocks to that. He's happy to BE the bad guy, just doesn't want to LOOK it.
"He's said he can wait til after Christmas then make him pay it off but I don't think this is fair. I'd rather resolve the problem now."
You're right to want to resolve it now. But really, it should have been resolved THEN. I mean, waiting for a 13 year old to offer? No he wasn't! Had your son thought to offer he'd have offered THEN. Your DP was not 'waiting', that's a bare-faced lie. He's peeved that your son is getting something and he wants to piss on your son's chips. Your DP needs to be the adult here and stop mewling that he wants a scratch on his bike expensively repaired before anyone else gets anything because he's the important person in this household and wants everyone to know it<rolls eyes>!
Side question - did your DP care enough about this scratch to instigate any repairs himself? Given it's been a few weeks since it happened? If not - then he's not really worried about the scratch.
"I feel like he begrudges my son of having anything good but at the same time I know my son needs to pull his finger out!"
Yes he does begrudge your son. And all 13 year olds need to pull their finger out
. But the fact remains that this was justcarelessness on your son's part, not deliberate - it was an accident. 13 year olds are growing and that affects their proprioception (awareness of where their body is in space) which can make them a bit clumsy. They overreach/underreach because their brain thinks their arms are still the same length they were last week. Frankly, they need to be cut a bit of slack on this.
"My partner has a son aged 17 and if he damaged something no questions would be asked. He gets anything he askes for and does nothing to help or to earn money (he doesn't live with us)"
I would be pointing this out to my partner. Forcibly. Because your partner is being a fucking hypocrite here, and I cannot abide hypocrisy. And for this reason alone, I'd be telling my partner that NO, my son will not be paying for a scratch that happened accidentally, and YES, my son will be getting this second-hand PC. And that I expect he, my partner should have a damn good think about how he treats my son and he'd better buck his ideas up considerably, because this picking on my son when he treats his own son so differently will no longer be tolerated.