I would ask the vet to do a Fecal Panel - IDEXX Real PCR Diarrhea Panel is the one our rescue vet uses for mystery diarrhea. It's a test that has to be sent out to a lab. It's fairly new (it actually tests for pathogen DNA), so it's possible some vets might not know about it.
I would not throw antibiotics at this not knowing what's there (Tylan is an antibiotic), as that's what leads to antibiotic resistance -- don't stop them without asking your vet, but I would call and have a phone convo about whether you might be creating something resistant by just guessing.
This panel tests up to 18 different pathogens that can't be seen on a fecal float slide. https://www.idexx.com/en/veterinary/...realpcr-tests/
We just had a "leftover litter" of rescue pups dumped at a shelter by a breeder who couldn't sell them all. They needed to have this panel run after they kept cycling through diarrhea -- they had THREE different pathogens (a virus and three bacteria), in addition to giardia. Any one of those bugs would have explained the diarrhea, but they had a mess of multiple stuff going on. If we'd only treated the giardia, we'd have had infectious pups spreading pretty nasty stuff and frustrated people wondering why they could never get firm poop out of them.
Giardia and parvo are both on the list of things it tests for, along with clostridum, camphylo, c. diff, and many other things, so you don't have to run separate tests any more (though you still can, if you prefer).
It costs around $200 and takes about 3 days. Most vets work with IDEXX, so your vet should be able to get you an exact price with a phone call.