Is there a test I can do to find out my mares parents?
Someone selling a horse can often get a higher price if the horse has papers. The seller can promise papers even if there aren't any. The words, "papers were to be sent with the mare," tell me that there were no papers. I think that the seller showed your family papers for some other horse, but you'll never be able to find out for sure, because you cannot inspect the papers. The answer from the wise person with the asterisk name is correct. Genetic testing is of no help for your situation.
If you didn't get the papers, your mare is no longer considered "registered" and by the way this is one of the most common scams in the horse world. Chances are, the seller will never send you those papers or if they do, they will be for a completely different horse. In the end, if you love her and she's right for you, registration doesn't matter. Just be aware that this missing papers thing is a total scam. With over 100,000 unwanted horses being shipped (in horrific conditions) to slaughter in Canada & Mexico, why would you want to create one more just because babies are cute? Do the right thing and don't breed.
That's not how DNA testing works. You can prove or disprove whether a specific horse is her sire or dam, if you have a genetic sample from them. Each horse checked would be a separate test and you would pay a fee per test. Essentially, you need to have a good idea what specific horses are the sire and dam before testing. I would not breed a horse of unknown pedigree. You have no way to research possible genetic health conditions within the lines.
There is DNA testing for horses at about $150. It will only tell the horses ancestoral breeds. A friend of mine did it to a raggedy little wearling she bought at auction. It turns out to have alot of spanish blood...
Judy and the Beast
If your horse were Arabian, yes you could confirm the parentage if it was known or suspected to be a few specific individuals. The AHA has been typing horses for some time now (for a large fee of course) and keeps these records. https://thehorse.com/15389/arabian-registry-to-require-dna-testing/ AQHA has similar requirements. Not sure of APHA... However, many studs are double registered so it may be possible. My good friend just got her unregistered Arabian registered after a lengthy search to "find"/find out his true parents. She had to provide the names and the AHA confirmed it by comparing his results to those in their database. This only works if you know or think you know the sire/dam.
If you don't have the papers, there is no proof she was from those bloodlines. The owners could use that set of papers for whatever animal they choose, but never send them along. Choose a sire you feel will compliment her confirmation. You cannot prove her lineage without docs. You might even consider looking into what crosses are popular, such as a good Arabian, Morgan, or pony sire. Look on horse sites for these. Paint cross can be shown in Pinto classes if the foal has color. Arabian WITH papers from sire can show 1/2 Arabian. Pony 13-14hh might be good. BUT consult vet for safety of mare before breeding, esp if older. Better to have a well-loved horse than problems. Full vet check before breeding. Of course, you could to go back to the seller to request these documents, but at this point they may not even acknowledge you, the sale, etc. This is, unfortunately, how many dishonest horse traders operate. Be appreciative of what you have, if she's been a healthy steed you love.