The incident happened Sunday afternoon in a residential area in Falkoping, central Sweden, the Swedish news agency TT reported.
The victim, a 27-year-old man, was barbecuing in his backyard when his neighbor, a 28-year-old man, showed up.
The suspect allegedly proceeded to attack the victim with the ax, police said.
"He hit him in the back with an ax," duty commander Johan Ljung of the Vastra Gotaland County police told the Aftonbladet newspaper.
After the victim fell to the ground, the suspect's father allegedly kicked the 27-year-old several times, police said.
"It was pandemonium. They were screaming and hitting," one witness told the Aftonbladet newspaper.
The victim was brought to a local hospital with injuries to his back and face.
Ljung said the altercation was likely sparked over opposing views in the neighborhood on whether grilling is acceptable.
"This family, who is known to police, clearly didn't think it was appropriate to grill," Ljung told the TT news agency.
The suspect fled the scene and had not been arrested as of Sunday night.