Parker, 33, is alleged to have stolen a purse and its contents belonging to Pauline Healey, the grandmother of 14-year-old Sorrell Leczkowski, from Leeds, who was killed in the attack on May 22
He is also said to have taken the mobile phone of a teenage girl who cannot be named for legal reasons.
The defendant made his remarks as he was led from the dock after entering formal not guilty pleas at Manchester Magistrates' Court to the charges of theft.
Following the attack, rough sleeper Parker received global acclaim after he described witnessing the effects of the blast and tending to the injured.
Speaking at the time, he said: "It knocked me to the floor and then I got up and, instead of running away, my gut instinct was to run back and try and help.
"There was people lying on the floor everywhere."
He told how he had wrapped an injured girl in a T-shirt and cradled a dying woman in his arms in the aftermath of suicide bomber Salman Abedi killing himself and 22 others.
On Wednesday, Ben Southam, prosecuting, said it was alleged that Parker took Mrs Healey's purse, containing bank cards, from a handbag as she lay stricken on the ground and also stole another victim's phone.
He added it was clear the defendant provided "some limited assistance" to people injured at the entrance to the venue's foyer but it was the Crown's case that he "equally" took the opportunity to commit the thefts in the immediate aftermath of the atrocity.
Mrs Healey attended the Ariana Grande concert with Sorrell and Sorrell's mother, Samantha.
She later underwent 15 hours of surgery to remove shrapnel from her body and also suffered multiple compound fractures to her arms and legs, while Sorrell's mother was also seriously injured.
Sorrell, who was a pupil at Allerton High School in Leeds, was hoping to be an architect and wanted to study at Columbia University in New York.
On the day of her funeral, her family said: "Sorrell was only 14, but she was our rock, she kept us all grounded.
"She was such a clever, talented, creative girl, there was nothing she couldn't do."
District Judge John Temperley said the case was too serious to be dealt with within his jurisdiction and must be heard in a Crown Court.
Parker, who gave the court an address in Woodlands Road, Crumpsall, Manchester, was remanded in custody until his next hearing at Manchester Crown Court on September 13.