Understanding the functional consequences of synaptic heterogeneity within neuronal populations is a critical challenge in neuroscience. Recent connectomic reconstructions, such as the complete fly brain connectome, provide key insights into the structural constraints on neural networks. One prominent observation from these datasets is the variability in synaptic connection strengths among functionally similar neurons, where certain neurons exhibit significantly more synapses than others. However, the role of this heterogeneity in neural function remains unclear. To address this, we present a statistical mechanics theory of parallel perceptrons, each performing independent classification tasks while constrained by a shared global distribution of synaptic weights. Our model reveals that synaptic heterogeneity naturally emerges due to inherent degeneracies in the solution space, predicting that neurons with higher in-degrees deviate more from a Gaussian weight distribution. These predictions align with empirical data from the Drosophila mushroom body, where the presynaptic weight distributions of Kenyon Cells vary according to their in-degrees. By applying our theory to the connectome data, we find that the family of heterogeneous weight distributions from Antennal Lobe Projection Neurons to Kenyon Cells is well-predicted by the model. This suggests that synaptic heterogeneity is a consequence of resource allocation under global statistical constraints: neurons with lower in-degrees receive more resources, while those with higher in-degrees compensate by developing non-Gaussian weight distributions. Our framework offers a novel explanation for the emergence of heterogeneity in neural networks, driven entirely by global synaptic statistics and connectivity patterns.