We consider communication over heterogeneous parallel channels, where a transmitter is connected to two users via two parallel channels: (1) a MISO broadcast channel (BC), and (2) a noiseless rate-limited multicast channel. We characterize the optimal degrees of freedom (DoF) region of this setting when the transmitter has delayed channel state information (CSIT) regarding the MISO BC. Our results show that jointly coding over the two channels can strictly outperform simple channel aggregation (or channel separation) and can even achieve the same performance as with instantaneous CSIT when the CSIT on the MISO BC is completely stale; this occurs when the multicast rate of the second channel is larger than a certain threshold, in the DoF sense. The main idea to achieve full DoF with completely stale CSIT is to send information over the MISO BC at a rate above its capacity and use the second channel to send additional side information to allow for reliable decoding at both receivers.