With the advances in optical technology, the span of a broadband access network using passive optical network (PON) technology can be increased from today's standard of 20 km to 100 km or higher, and thereby serve a lot more users. Such an extended-reach PON is known as SuperPON in the literature, and we call it a long-reach PON (LR-PON). A major challenge in LR-PON is that the propagation delay (for data as well as control signals) between the telecom central office (CO) and the end user is increased by a very significant amount. Now, traditional PON algorithms for scheduling the upstream transmission, such as dynamic bandwidth allocation (DBA) algorithms, may not be sufficient; actually, they may lead to degraded performance because of the long delay of the CO-to- Users "control loop." This challenge motivates us to propose and study a multi-thread polling algorithm to effectively and fairly distribute the upstream bandwidth dynamically. This algorithm exploits the benefits of having multiple polling processes running simultaneously and enabling users to send bandwidth requests before receiving acknowledgement from the CO. We compare the proposed algorithm with traditional DBA, and show its advantage on average packet delay. We then analyze and optimize key parameters of the algorithm, such as initiating and tuning multiple threads, inter-thread scheduling, and fairness among users. Numerical results demonstrate the algorithm's advantage to decrease the average packet delay and improve network throughput under varying offered loads.