So there's an interesting quirk on this 1890 map - https://maps.nls.uk/view/229946781 with "Back Cable Street" appearing behind buildings on "Parliament Street" on what is now the car park for the Parliament Street Retail Park.

That aside, all the other evidence I have is that Parliament Street runs as far as the North Road junction on both sides.

The key point is this bus lane order from 2019 - https://council.lancashire.gov.uk/documents/s155424/Appendix%20A.pdf
- While it has maps that we can't use, it does have a textual description which presumably we can use which talks about a bus lane starting on Parliament Street 105 metres south west of the Centreline of Bulk Road, to the left turn filter at its junction with Greyhound Bridge Road. That to me is enough evidence that https://www.openstreetmap.org/way/96055314 is not Cable Street and is in fact Parliament Street.

Updated in https://www.openstreetmap.org/changeset/170353883