Convolutional neural networks are an important category of deep learning, currently facing the limitations of electrical frequency and memory access time in massive data processing. Optical computing has been demonstrated to enable significant improvements in terms of processing speeds and energy efficiency. However, most present optical computing schemes are hardly scalable since the number of optical elements typically increases quadratically with the computational matrix size. Here, a compact on-chip optical convolutional processing unit is fabricated on a low-loss silicon nitride platform to demonstrate its capability for large-scale integration. Three 2 × 2 correlated real-valued kernels are made of two multimode interference cells and four phase shifters to perform parallel convolution operations. Although the convolution kernels are interrelated, ten-class classification of handwritten digits from the MNIST database is experimentally demonstrated. The linear scalability of the proposed design with respect to computational size translates into a solid potential for large-scale integration.