Exact way of concatanetion is not clear but it is derived from roots of “Per” and “Iz”. First root has meaning just like it is in English. (Turkish word for one is “Bir”/"Bi". Associated with Pi.). Second root means trace in Turkish. Because of catwalking, cats under certain circumstances put their next step into the exact place of previous one, cats are named “pars”. Simply because they put their foot to same place twice. Like “Per Trace”.
Altough in modern Turkish cats are called “kedi” similar to common name. When people call a random cat they don't say “kedi”. They say “pisi pisi”. It is also a name for cats in other Turkic dialects than Turkey. “pisi” is not much different than “pars”.
So in ancient times they were referring traces of cats as “pars”. (Video of a cat catwalking on desert)
(Meanings of Persian word Perhiz depicts qualities of cheetahs. Used in Turkish very similarly. Arabic word Bariz, means obvious, evident. Used in Turkish with same meaning. European name Boris probably has similar origin although it is not possible to draw lines for any of these. Obviously "Persia" is also connected.)
(Turkish word for peace is “Barış”. First root should be the same. Second is also same or related or maybe something like infliction. Actually English word like peace, piece also sounds similar. Not going into things like these much because it is not possible to draw lines. However the word for one is “Bir” and Bar or Par not different roots. And peace and piece are good examples that can relate. Peace actually sounds very similar to word for calling cats “Pisi”. Which is sometimes name of cats in some dialects.)
(English "for" is also related to first root.)
(Wiktionary pages: for Old Persian, for Hebrew)
“İz” is probably most important root. Placed to everywhere by Uzers. And actually part of many important special names like Islam. Almost all Greek words ends with “Us” or “Is”. This wasn't probably the reality back then. (This is very complicated. There is even a letter introduced around 9th century. “ς”. It is considered positional variant of Sigma but overall very suspicious. And actually there is a letter in Quran that its shape slightly changed and “ς” sign put on top around 9th century. It is the standard after 10th century and in modern mushafs but didn't exists until 9th century. Interestingly at the same time both Greek alphabet introduced with this letter and shape changes made to letter from Quran. Both are final letters.) (Ancient Greek was something like Turkish. Letters corresponds to roots. Modern Greek is extension of ancient letters with marks and mapping one letter to multiple one just as it is Hebrew and Arabic.)
Various sources claims a connection between name of Persia derives from word for cat. And this claims seems like even exists independent of Turkish word "pars" which still used for some type of leopard. Obviously it is hard to understand if this was the case how this is forgotten. But there are many historical arguments to support the connection. I can also suggest another speculative one. If cats were named it because they were stepping same spot twice. Maybe people who travel abroad were referring there as the same because they were going through it in the both outbound and inbound trip. This may sounds ridiculous at first but in past travels were mostly done for purpose of trading and obviously traders would prefer different routes to explore different markets. Just a speculative idea. The area is extremely mountainous. And it seems like there weren't alternative routes through there. And often mentioned as a hub or spine of silk road. (El Paso actually was named with this logic. At least in some instances.)