I have just started working with a not for profit organization to help get the accounting in better shape. I have only a rudimentary knowledge of accounting but have the time and educational experience to come up to speed eventually.
The paid workers in our organization are actually employees of another organization(#2). All of our money is held and checks issued by a 3rd organization (#3). Bi-weekly we submit time cards to organization #2 and that organization issues paychecks. Organization #3 is billed for the total worker wage expense amount 10-15 days after the end of the month and the amount owed is paid from our account. We would like to keep track of our liability for wages, pension, FICA in separate accounts. I am unsure as to possible account titles which could or should be used and to the various DR/CR for each step. Any advice?

Who submits a tax return for you? Org# 2?
What method of accounting are you presently using?
If allowed by your parent organization, I would keep it cash basis. Income and expense. When you reimburse for payroll, that's a fee. Payroll breakdowns are for your eyes, since you have no employees. If you have to be accrual and have to make payroll gibe, I would discuss the 3 -5 week lag with your parent org.