Yes we do, it's a cyclooxygenase inhibitor. E.g. from the abstract of [1]:

"The mode of action of paracetamol has been uncertain, but it is now generally accepted that it inhibits COX-1 and COX-2 through metabolism by the peroxidase function of these isoenzymes."

Furthermore, the active ingredient in Tylenol is a very simple molecule from a synthetic perspective, and it was developed while trying to overcome the toxicity of an even simpler molecule called acetanilide. Even though it was developed 150 years ago, there was a systematic understanding of organic chemistry and the interaction with the human body already at that time.

[1] https://link.springer.com/article/10.1007/s10787-013-0172-x