Police are investigating the incident, which happened outside the Tesco Express on Charminster Road and involved the female pedestrian in her 60s, a black Honda NSC 110 Scooter, and a parked BMW.
Officers were called to the scene following the crash at about 8.05pm on Sunday, January 25.
The rider of the motorbike fled the scene and a 16-year-old boy from Poole has now been arrested on suspicion of driving offences and released on police bail.
Police Sergeant Paul Hind, of the Roads Policing Team, said: “I am appealing to anyone who was in the area and saw what happened to please come forward.
“Also, I am keen to speak with any motorists who were travelling along Charminster Road and may have captured anything of relevance on dashcam footage.”
Anyone with information is asked to contact Dorset Police online, via email at scit@dorset.pnn.police.uk or by calling 101, quoting occurrence number 55260011600.
Alternatively, independent charity Crimestoppers can be contacted anonymously online using its website or by calling Freephone 0800 555 111.