Wake-Up Radios is an emerging technology, aiming at pushing forward the frontiers of energy efficiency without trading it off for latency nor reliability. Extending the lifetime of the nodes as much as possible is one of the main goals in Multi-hop Wireless Sensor Networks. The Routing Protocol for Low Power and Lossy Networks (RPL) is commonly used in these applications. However, there is still an open problem in its design when it comes to achieving both stability and efficient routing at the same time. In this article, we present Load Balancing Parent Selection (LoBaPS), an algorithm to select opportunistically the next hop, based on RPL. It capitalizes on the Wake-Up Radio and its always-on feature, as well as its Ultra-Low Power consumption. We compare the performance of LoBaPS with that of W-MAC, a reference protocol that uses Wake-Up Radio and supports RPL in its traditional way. The results are obtained through simulations in COOJA for a network of nodes running ContikiOS, and show that the lifetime can be improved up to 55%, while the Packet Delivery Ratio (PDR) can raise a maximum of 20%, keeping a reasonable level of latency. In addition, the network is more robust to node shutdowns and requires less control overhead.