Stool tests are remarkably inaccurate... less than 20%. The parasites don't always shed eggs/body parts. When they do, it's not uniformly mixed in your stool and no guarantee that the bit of poop you provide them will contain anything. Then, there is the skill of the lab technician.
Elisa tests, according to Mattk3, are better, but they are specific to each species. And, even then, there is a margin of error in interpreting results.
I've been tested 3-4 times at various stages of multiple parasite infections, and none of the stool tests came back positive. It was an Ayurvedic doctor feeling my pulses that correctly identified my hookworm infection. The few conventional doctors (even at a tropical disease hospital in Thailand) were worthless and wouldn't go beyond the stool test.
One of the easiest self-tests is a dose of Ivermectin or Pyrantel and see what comes out. It won't pick up protozoans, but you'll see worms and flukes.