Are Newry council car parks and the Newry on-street controlled zone enforced by the same authority?

Not necessarily.

DfI is responsible for paid on-street controlled parking in Newry, while local councils manage their own off-street car parks. Both may use a Penalty Charge Notice system, but the operator, payment channel and formal contact details can differ. When disputing a ticket, look at the issuing body printed on that notice rather than choosing an appeal link based only on the fact that you were in Newry. Keep the bay or car-park location and photographs of signs.