Avon and Somerset Police said this morning that Jordan Walters, 24, and Emma Jane Horseman, 23, had been charged in relation to the incident.
They have been remanded in custody to appear before Bristol Magistrates’ Court on Monday 4 July.
In a statement, the boy’s parents appealed for privacy.
Amy Allen and Edward Studley said: “We would like to thank everyone for their continuing support at this time.
“Harry is still very poorly and we would reiterate that our request for privacy be respected as we and our family come to terms with what has happened.”
Detectives said that the baby boy had been “shot by a pellet discharged from an air weapon”.
In a statement, Avon and Somerset Police said: “We‘ve charged two people following an incident in Hartcliffe in which a one-year-old boy was seriously injured.
“Harry Studley remains in a critical condition in Bristol Children’s Hospital with a serious head injury.”
Emergency crews were called to a flat in Bishport Avenue, in the city’s Hartcliffe district, at 4.10pm on Friday, after neighbours reportedly said that they heard the child “screaming and screaming”.
The Bristol Post quoted an unnamed father, who witnessed the aftermath, as saying: “I couldn’t sleep last night because I just couldn’t get this picture out of my head.
“The mother was running behind covered in blood.”