Diffusion MRI (dMRI) tractography enables in vivo mapping of brain structural connections, but traditional connectome generation is time-consuming and requires gray matter parcellation, posing challenges for large-scale studies. We introduce DeepMultiConnectome, a deep-learning model that predicts structural connectomes directly from tractography, bypassing the need for gray matter parcellation while supporting multiple parcellation schemes. Using a point-cloud-based neural network with multi-task learning, the model classifies streamlines according to their connected regions across two parcellation schemes, sharing a learned representation. We train and validate DeepMultiConnectome on tractography from the Human Connectome Project Young Adult dataset ($n = 1000$), labeled with an 84 and 164 region gray matter parcellation scheme. DeepMultiConnectome predicts multiple structural connectomes from a whole-brain tractogram containing 3 million streamlines in approximately 40 seconds. DeepMultiConnectome is evaluated by comparing predicted connectomes with traditional connectomes generated using the conventional method of labeling streamlines using a gray matter parcellation. The predicted connectomes are highly correlated with traditionally generated connectomes ($r = 0.992$ for an 84-region scheme; $r = 0.986$ for a 164-region scheme) and largely preserve network properties. A test-retest analysis of DeepMultiConnectome demonstrates reproducibility comparable to traditionally generated connectomes. The predicted connectomes perform similarly to traditionally generated connectomes in predicting age and cognitive function. Overall, DeepMultiConnectome provides a scalable, fast model for generating subject-specific connectomes across multiple parcellation schemes.