In one view, the etymological root of Sparta isn't secure. So the answer to your question is maybe! But it's kind of unlikely that the Spartans were named for being frugal, if you think about, and it doesn't seem as if that's what you meant. In that sense, the root of a word is what it has meant for the longest time--otherwise the root of all words would be "ook!" or "uga-uga", which is a matter of debate. Most people do not draw the comparison to "sparse", where as we lexicalize "Spartan" as a proper name without any deeper sense. I suspect that's one aspect of your question. So, you are smarter than them in this respect, but perhaps too smart, because, until further notice, the answer has to be: No! This is in principle the extend of how far an answer has to go on ELU.
Further: @Kris' remark is notable, both words might be connected through a common root, but how big and indirect the distance is between them would be decisive for calling them related at all. So the answer is: Yes?!?. But it's far from certain.
@vectory alleged a comparison to spared. This is indeed one semantic connotation of venerable age for Latin servus "slave", at least, and possibly for slave, if from a sense "to get the spoils of war". Even if those might not be the correct etymologies, it's not said that the ancients necessarily knew that. So, my answer is: I guess so. But I wouldn't pay it much attention. But the point is, that's quite a different sense of to spare than we would think of today.
We can say at least that -se and -ta do, obviously, I guess, not reflect the same morpheme.