I'm looking for a story (novelette, short, etc) I've read in an anthology, about a human running from pursuit who lands in a planet and enters in a cave where something falls on his shoulders and he faints. Upon awakening he discovers he's bonded to an alien, still on his shoulders, that gives him mind powers.
I wonder if you are remembering the novelette Pard by F Paul Wilson.
The reason I say this is that your description sounds almost exactly like the opening chapter of Healer as suggested in Fuzzyboots' answer, and the novel Healer was adapted from the novelette Pard published in Analog in 1972. Specifically Pard evolved into the first chapter of Healer.
The scene in Pard where the protagonist Steven Dalt is attacked by the alaret is almost exactly the same as in the novel. He has landed on the planet Kwashi and is attacked by tribesmen. He hides in a cave to escape them, and an alaret lands on his head and he faints from the pain. When he comes to he finds he has the alien as a mental symbiote. The only difference from your description is that the body of the alaret dies and falls off after it has infected Dalt. It does not remain on his shoulders.
If you read Pard in an anthology somewhere that would fit your description but without the huge scope of Healer.
Could it be F. Paul Wilson's Healer?
In The Healer, a human, Steven Dalt, travelling to another planet finds himself in a cave, where an alien creature fuses to him forming a symbiotic relationship. Steven was an ordinary person with nothing fantastic about him, but once the alien attaches to him, he becomes this superhuman immortal. The alien creature has incredible knowledge and skills that Steven does not. On the other hand, Steven has the body the alien did not. He becomes The Healer, capable of healing almost anything, including himself. The story goes on for centuries across galaxies and eventually comes to a conclusion.
Wilson wrote a newer book with the same alien symbiote: Double Threat, June 2021.