the same party; the party in question
Refers to a political party previously mentioned or understood from context. Common in news reports.
The candidate from the same party won the election.
the said party; that party
Used anaphorically to refer back to a political party already mentioned, similar to 'the aforementioned party'.
The said party announced a new policy.
The party announced a new policy next month.
A lawmaker from the (same) party held a press conference.
A conflict of opinion is occurring within the party.