I wouldn't.
You have no real way to tell exactly what they've changed. Some schools are excessively nosy or controlling.
And even if the district is being respectful of your privacy, they could have a rogue admin in their ranks.
Others have been bitten.
There [have been lawsuits][1] because of blatant misconduct before. They have alternatives, so administrative access should not be necessary.
How should they do it?
Cloud-based software requires no installation. As long as you have a modern OS and web browser, you're ready to go. While I dislike cloud apps in a number of scenarios, it's perfect for bring-your-own-device (BYOD) scenarios. Obviously, they did not choose this if they're asking for admin rights. You might suggest it to them.
With volume-licensed software, they should able to provide a product key or setup a license server on their network. (The stuff that requires license servers is more common for university-level applications, but I've heard of it in technically-oriented college prep schools, too.)
What would I do?
I would install the applications myself. It doesn't take a lot of time, and typically they don't change over the course of the year.
Certificates can be installed very easily on Windows, but I'd have to see them first before I could say whether or not that's a good idea. [1]: https://en.wikipedia.org/wiki/Robbins_v._Lower_Merion_School_District