Yes, but I think we cheated a bit, because I already had DS1 when we got together (he was 2) and got married (he was 5), so we already had that sense of what we would be like as co-parents and DS2 has just sort of magnified the luuurve in a really sickening way (sorry :o) and also we were quite lucky in that we seem to match well on some important points.
I think you need some stuff to just be there, you can't force it, if it's not there it just won't work. Shared/compatible worldview and values. Mutual respect. Emotional healthiness aka not being abusive/controlling/overly insecure. Or just owning your own issues, not making them your partner's problem - general relationship health stuff, not necessarily perfect mental health.
You've got to both be proactive about responsibility and see it as your thing - you can't be like "Not my job" because that is essentially saying "It's your job, that's beneath me" which is a huge fuck you to your partner. Goes for housework, cooking, household admin, financial stability, pet care, anything like that. Doesn't need to be a strict 50/50 split, actually I think a flexible ebb and flow approach to this kind of thing is more of an indicator of it being fair because it implies you're not having to constantly police who is doing more/less.
About what you can consciously do -
Never be in competition. Don't play tiredness bingo and don't wish that the baby will play up for them. You are on the same team so do what you can to make it easier for your team mate. And ask when you need them to make things easier or relieve you.
When you disagree, it's not about winning/who is right. Try and each distil your own position down into what the most important parts are and see if there's any way you can get the two to mesh. E.g. at 5 months you are co-sleeping and your partner wants to sleep train. You can argue about which approach is beneficial endlessly in circles or you can say - it's important to me that we don't leave the baby to cry, and also I don't have the energy to implement some kind of plan in the middle of the night, he says - it's important that we get some more sleep, the solution might be separate bedrooms for a bit/alternate lie ins at the weekend, some kind of sleep plan that doesn't involve crying, a promise to reassess if no improvement by 7 months, etc. But you can easily get sucked into "THIS is the way we solve it" and butt heads.
Remember the respect that you have for your partner aka why you married them/had a child with them. So don't assume that you're right and they are automatically wrong/stupid/don't understand - their point of view is probably relevant too. I see a lot of people sniping at their partner "not getting basic things" whether it's that he doesn't follow the sterilisation guidelines you do (guilty) or that he can't pick a matching outfit. You've essentially got to trust them to be competent. Which means respecting it when they make decisions differently to you, and also not giving them a string of instructions all the time, let them make mistakes, it's fine. Don't hole yourself up on the net also getting pulled into elaborate belief systems about various aspects of childcare. If you feel this is happening then either step away or invite your partner to read some of the stuff you are reading and see what their take on it is as a sort of "sanity check" - because you respect their opinion. NB it doesn't mean you can never tell them "I'd prefer it if you didn't do that" but it does mean don't assume that you have total veto - be prepared to see it from their point of view. Prob a good idea to get into a habit of checking with each other if you're both OK with new things.
Be flexible - parenting is impossible to imagine the reality - if something isn't working you have to be open to changing it. It's not a failure. Talk about short-mid-long term plans and various options for them, regularly.
Actively defy sexist stereotypes - parenting is really sexist - it's a good idea to try just as a thought experiment specifically imagining and discussing how opposite-sex roles would work for both of you e.g. dad being stay-at-home parent, mum being breadwinner, or mum's work being sacrament while dad drops to flexi time, deals with pick ups and drop offs, sick days, assemblies, or works part time. You don't have to do it, but it really helps to discuss it.
IMO if you find you're in a relationship (before kids) where any of these seem impossible because of how your partner is you will not be happy after kids. If you're always in competition (even jokingly), if you/they aren't prepared to compromise, if you don't actually respect them or believe them to be competent, if they are inflexible, if they are strongly attached to gender roles. These are red flags and make it really difficult to co-parent happily.