It's almost never linemen's parents who are the issue, fwiw. (lone exception was a DE/athletic freak and his no nonsense single mother).
Thinking it through, this would be my suggestion:
Winter (nov-dec-jan) - LP and CYO basketball. I'm assuming the basketball is recreational and not demanding.
Spring (feb-mar-april) - Track throws (preferable over LAX - why? because of all the damned running in LAX!). Either sport, LP can morph into something like in this article -
The Novice Program for High School Athletes
Summer - May - Just LP. June - Continue to LP, but attend a positional camp. July - condition for training camp. Unfortunately, football training camps - two a days, conditioning tests, etc. are not designed to make players better. They are designed to be a crash course in team formation. In July, he'll need to make sure that if he has a stupid ass conditioning test he prepares for it, no matter how dumb. Otherwise, positional starts, short (10-30yd) hill sprints (in cleats), and up-downs will be the most valuable conditioning he can do.
The additional jump rope is not necessary. Here's why. Basically, it's not that it isn't a good exercise, it is. It is not a necessary one because most coaches lack a useful way to describe what they actually want.
In coach speak "Lighter on their feet" for line really means "Be in the right position, at the right time, and able to apply force and leverage with violence."
It's not about foot speed, per se. It's about not wasting steps, not losing technique, and not blowing assignments. Seriously - if he knows what he is supposed to do, it will look like he got faster feet. He would be much much better off, IMHO, going to a positional camp and learning the game along with the concurrent increases in his strength. If your son knows exactly what to do for every protection and his assignments on inside zone, power and counter (the three most common run plays that nearly everyone runs), I can almost guarantee you'll hear about how he's so much faster than last year. If you've got a scheme specific question or question on how to watch film, ping me direct.