"essentially attempting to create a new appeal against me." No. A new appeal against you cannot be created in this manner. You have no grounds to move to strike the respondent;s brief in the cross appeal because you are not a party to the cross-appeal. If this party wants to make the same arguments against the other defendant as they did against you that is perfectly legal.
See Cal. Rule of Court 8.108(g): "(g) Cross-appeal
(1) If an appellant timely appeals from a judgment or appealable order, the time for any other party to appeal from the same judgment or order is extended until 20 days after the superior court clerk serves notification of the first appeal."