The 25-year-old, from north London, is accused of inflicting grievous bodily harm on Mr Charlton-Weedy.
He has also been charged with possession of an offensive weapon and possessing cannabis.
The victims of the stabbing were taken to hospital but none of them are believed to be in a life-threatening condition.
Kishoree Kotecha-Pau, Parkinson’s lawyer, made no bail application on behalf of the defendant, but spoke to remind the court that he was arrested at his home rather than at the scene.
Aarya Mishra, prosecuting, said: “The defendant and another man were together cycling through London on a pedal cycle.
“They were together, trying to snatch phones from various victims’ hands. This led to what has been described as an incredibly violent series of events.
“The defendant tried to snatch a phone from a member of the public who cries out to other members of the public to try and detain the defendant and his associate.
“His associate produced a knife and slashed various members of the public. He then handed that to the defendant who slashed a member of the public’s face so gravely that he required 52 stitches.”
Magistrates remanded Parkinson into custody and he will next appear at Inner London Crown Court on November 8.