This is a big one. Everyone struggles with this and you will have to find what works for you. There are many ways to manage your time.
The early-bird option:
Debbie works full time, has three kids and a husband who works full time. She does triathlon to have something for her. Her “me time”. Because of all she has to manage and shuffle, kids to soccer and music practice, as well as her full time job, and getting dinner on the table, she decided to focus on the Olympic Distance triathlons. She has a husband who is happy to pick up the slack and is very supportive, but she doesn’t want to miss any of the kids’ activities or events. She concentrates her training time to early mornings. She is up at 5 am most mornings and out of the house. Tuesdays and Thursdays to she attends her Masters Swim class that starts at 6:30 am. On Wednesdays she jumps on the trainer in the basement for an hour, and on Fridays she hits the road for a run, often with her headlamp on and bright reflective clothing. On the weekends she does her long runs and bike rides. Then too, she generally gets off in the morning so that she can spend the rest of the day with her family.
While someone else watches the kids option:
Sally is a single mom of two and works full time. She was a college athlete and is training for her first half ironman. This training regimen has her training about 15 hours a week. Her long bike rides on the weekends last for about 3 hours. Because she can’t leave her children home alone early mornings and sneak out to train, she gets much of her training done during her lunch hour. She also can get a run in at the end of the day while they are at aftercare. On weekends, she plans play-dates for her two daughters and at times has hired a babysitter. Some weekends she has to do the long rides on the trainer watching a movie with her daughters. It can be tough at times but it means a great deal to her to compete at this distance and she is getting the training done. Her daughters are very proud of her and are always encouraging. That fact alone keeps her going.
It can be tricky, but there are options. Schedule your time before work, during your lunch hour, or late night trainer time when everyone is asleep. And plan your race distance based on what you can do.