Bit of a strawman argument, since said parents are currently (usually) paying into the system.
It's not a strawman. It's a valid analogy. In pretty much any school district I know of, anyone sending their children to public schools is getting more than they are paying for. That's the nature of making education a publicly financed service. It's the nature of pretty much any publicly financed service. It is a form of wealth redistribution and the same arguments apply.
And my concern is that it's a likely vehicle for expansion of government power if we allow this kind of intervention as a precedent.
The issue of testing welfare/food stamp recipients is more along the lines of making sure what we are paying for is being used properly.
Right. The same argument can be made for public education. Well-meaning citizens, interested in making sure that their tax dollars are being used wisely, might insist the surveillance cameras be placed in the bedrooms of all students - to ensure that they're doing their home work regularly and that they're not cheating. The point is, there all kinds of ways we might like to use government to control other people. But it's in all our interests to set strict limits on how far that goes.