Is it possible to use pi web parts 2013 sp1 (sharepoint 2010) with pi web services 2010 r3 installed on a different server?
Questions about Prerequisites should be answered by a products Release Notes. After briefly looking into PI Web Parts 2013 SP1 the Release Notes and PI Web Parts 2013 Administrator Guide my understanding is that PI Web Services is not a requirement but PI Web Parts 2013 provides the general ability to consume data from web services.
I conclude the issue you experience might be more related to establishing a remote connection to PI Web Services. Is this correct?