This will work definitely Try this
update VectorReplica.dbo.PARCELS
set PARCEL_ID = A.new_pid
from Ranieranneaxparcels A
where VectorReplica.dbo.PARCELS.PARCEL_ID = A.NEW_PID
or if output is 0 row affected than you need to consider where clause. It might be possible no any row is = to each other.