The answer to this question (in some Chinese courts, at least), is yes. A recent legal blogpost flagged a ruling in a labor case published in the Supreme People’s Court’s case database: Su Qiao v. the Taian (Shandong) Municipal Communist Party Disciplinary Inspection Commission. The author of the blog cited in support of his view Articles 48 of the 2012 Civil Procedure Law and Article 52 of the 2015 Civil Procedure Law Judicial Interpretation (concerning organizations that can be parties to civil litigation).
A subsequent (partial) database search revealed some other civil cases in which Communist Party organizations have appeared in variously as plaintiff, defendant, third party, and party against which the enforcement of an arbitral award was sought, including one decided by the Supreme People’s Court (Court):
General Office of the Jinan (Shandong) Communist Party Committee v. two individuals (leasing dispute,)(application to withdraw the lawsuit);
Three Gorges United Vocational University v.Chongqing Jianan Construction (Group) Ltd.Beibei District, Chongqing Muncipal Communist Party Committee School [Chongqing Jianan is listed as one of Chongqing’s top 100 companies in 2010], (construction dispute)(ruling by the Court in application for re-trial);
Beijing Forbidden City Film Co. Ltd. v. Jiangchuan County (Yunnan) Communist Party Committee Propaganda Bureau (enforcement of a Beijing Arbitration Commission arbitral award).