Masters in education during physics PhD
Seems to me like a long shot. There are a lot of obstacles. But first, few secondary schools require a doctorate to teach. So, what is your real goal? The doctorate? Secondary teaching? 

Next, you won't have much of a life if you try it. Think 80 hour weeks for a couple of years. A doctorate doesn't often get easier after you pass comps. 

Just the scheduling will be an issue. If you are taking courses in two departments, they won't coordinate class times to make it easy for you. 

But, perhaps, the biggest obstacle is fining an advisor who thinks it is a good idea. "Where is your commitment?" 

As for being competitive for admission, I think it might be a minus. Again "Where is your commitment?" 

Costs might be an issue or not. A doctoral student in the US normally has funding sufficient to live, but it normally comes with duties that take time. Tuition is normally waived. But it might not be for the other program. 

But, it may be possible (just) if you want to completely set aside all other concerns for a couple of years. Hard, though. 

Most secondary teachers with degrees in field, get their education credits over time through continuing education. It is the normal practice in the US. Finding time for a course or two in field-related pedagogy is a more modest goal, and more feasible. But still not essential, as few people actually do that in the US.