Okay there were many comments here... I'm a dad of 5 and 7 year old girls. They're now old enough so I start teaching them to code, and then I plan to take it from there some day (learning together). Besides that, I always have an hour or two each day to do what I want, when they sleep. In the first year with each kid it's hard to find any time, which is fine, but then it just gets easier every year.
Another way for you to keep learning might be to found your own company - just be careful to focus and not to neglect your family. With my first company, I was lacking focus and put too much unnecessary work into the company, to the cost of my family. Also, my co-founders didn't have kids.
The second time I founded a company with another dad, who understands flexible schedules. We're now a few moms and dads in the team. Sometimes I would go home early, be with my daughters, and catch up with work later. I feel I'm a lot more focussed than before having kids, and I definitely get a lot more done, in less time! It doesn't even compare. I don't have a choice anyways. My learning mostly happens on the job. When we get a new project, we sometimes decide to try a new technology - if it seems to add a benefit or save us time in the long term.
Ah yeah and of course, communicating with your partner is key if you want to shave off some time here and there!