In the current world pandemic situation, the contagious Novel Coronavirus Disease 2019 (COVID-19) has raised a real threat to human lives owing to infection on lung cells and human respiratory systems. It is a daunting task for the researchers to find suitable infection patterns on lung CT images for automated diagnosis of COVID-19. A novel integrated semi-supervised shallow neural network framework comprising a Parallel Quantum-Inspired Self-supervised Network (PQIS-Net) for automatic segmentation of lung CT images followed by Fully Connected (FC) layers, is proposed in this article. The proposed PQISNet model is aimed at providing fully automated segmentation of lung CT slices without incorporating pre-trained convolutional neural network based models. A parallel trinity of layered structure of quantum bits are interconnected using an N-connected second order neighborhood-based topology in the suggested PQIS-Net architecture for segmentation of lung CT slices with wide variations of local intensities. A random patch-based classification on PQIS-Net segmented slices is incorporated at the classification layers of the suggested semi-supervised shallow neural network framework. Intensive experiments have been conducted using three publicly available data sets, one for purely segmentation task and the other two for classification (COVID-19 diagnosis). The experimental outcome on segmentation of CT slices using self-supervised PQIS-Net and the diagnosis efficiency (Accuracy, Precision and AUC) of the integrated semi-supervised shallow framework is found to be promising. The proposed model is also found to be superior than the best state of the art techniques and pre-trained convolutional neural network-based models, specially in COVID-19 and Mycoplasma Pneumonia (MP) screening.