I would add to EmandM's excellent answer that 3 is old enough to begin to learn about other people's bodily boundaries, and in general that one does not touch others in a way (or at a time) that is unwanted. You can tell him, "I don't want to play this way right now. Please give me some space!" You can also redirect to a type of touch or a time that IS acceptable. "But I would love a hug" or " We can wrestle after dinner". Keep it friendly but firm when you let him know what the boundary is, and escalate as needed if he then chooses to ignore it.
You may have to enforce your boundaries with whatever disciple technique you commonly use until he begins to get the idea. (I don't mean to suggest anything too harsh, he's just seeking your love and attention-- but he should get the message that doing it in a way that is painful to you, and continuing after you have asked him to stop, is not acceptable in your family).
It is absolutely crucial, in my opinion, for kids to learn that they must respect other peoples' "no" when it comes to physical touch, and that means parents as well. If someone says stop, they should stop right away, and if they tell someone else to stop touching them, they should expect, even demand, the same. It's a cornerstone of successful social interaction as an older child and adult, but a hard lesson at first for a toddler, who may still be coming to terms with the idea that other people have their own needs/wants/feelings.
Overall, he sounds like he's mostly bored and seeks attention and connection. Being home most of the time and entertaining himself while parents give their focused attention to work or older kids is not an easy thing for a little child, even with other kids to play with. He might respond well to being given some 'homework' in the form of puzzles, coloring, etc while you are teaching the others, or to having more one-on-one time with adults.