Here are some suggestions.
Read while waiting for a bus or while riding a bus.
Many books are available as audio tapes. Listen to these while driving a bicycle or car.
Carry a book in your pocket and read it at any odd moment, such as while waiting in line at the grocery store.
Read while eating lunch at work, and at coffee break.
Bring work-related reading material to meetings. Read it before the meeting starts, and maybe divide your attention between it and the meeting if the meeting strays onto topics that don't directly concern you.
Be selective about what you read. Skim some things, ignore some things, and choose a small amount to read with full attention. Learn to find the most important parts of an article.
Most parents sleep fewer hours than their children, so you may be able to read while your children sleep. Just before bedtime is often a good time to read. However, many adults need more sleep than they think (see the book "Sleep Thieves"), so it's not worthwhile cutting into your own valuable sleeping time.
Read for 2 minutes or 5 minutes at a time while standing in the kitchen. Children will let you get away with this much more easily than if you sit down to read. Stop when the time is up: this teaches the children, subliminally, that they don't need to interrupt you.
Make a deal with the children: spend 10 minutes reading aloud to them from a book they choose (or 10 minutes for each child), then 10 minutes of you reading by yourself while the children play quietly. (Or you get your 10 minutes first.)
Teach your children to read, and have a reading time each evening when they read to themselves and you read to yourself. At the end you could all talk to each other about what you've read. You could try to briefly express something you've just learned in terms that are understandable and interesting to a child, maybe in a single sentence; that could be good practice in communication for you, help you understand and remember what you've just read, and help the children learn about the world of work. (Or could help you realize that what you're reading is a waste of time.) And the children could have fun telling you about the stories they've just read, or reading them again aloud to you if they feel like it. Here's my page about teaching children to read: http://web.ncf.ca/an588/phonics.txt
Take the children to a playground and spend part of the time reading while they play.
Read for hours while flying in an airplane, riding a train etc.
Decide not to read. Maybe you have more important things to do than read, these days. Spending time with the children is very important. Well, teaching them to respect your quiet time while you read for 10 minutes every evening can also be important: that's teaching them patience etc. You can do some of each.
Discuss stuff with your co-workers or other people. Get them to tell you the most interesting information from what they've read, so you don't need to read it yourself. Get them to recommend what are the most important things to read.
Go ahead and read in your office if it's part of your work. The question is not whether anyone else does that; the question is whether it's the most worthwhile
thing you can do for your job right then.
These are all just suggestions. You can choose any that make sense to you.
Inability is an abstract thing involving comparison with alternate universes; it cannot be experienced.