In my opinion no. You cannot hold two separate non-immigrant visas/statuses concurrently. You can have two H-1B's with separate employers working part time or full time for each of them but not two separate non immigrant visa statutes like an H-1B and an R-1 concurrently.
You can have concurrent H-1B jobs, but not two visa statuses at once.
Mr. Shusterman's (former INS Trial Attorney, 1976-82) response to your question is general in nature, as not all the facts are known to him. You should retain an attorney experienced in immigration law to review all the facts in your case in order to receive advice specific to your case. Mr. Shusterman's statement above does not create an attorney/client relationship.