Josephson parametric amplifier (JPA) engineering is a significant component in the quantum two-mode squeezed radar (QTMS), to enhance, for instance, radar performance and the detection range or bandwidth. In this study, we apply quantum theory to a research domain focusing on the simulation of QTMS radar. We simulate a proposal of using engineered JPA (EJPA) to enhance the performance of a QTMS radar. We define the signal-to-noise ratio (SNR) and detection range equations of the QTMS radar. The engineered JPA leads to a remarkable improvement of the quantum radar performance, i.e. a large enhancement in SNR of about 6 dB more than the conventional QTMS radar (with respect to the latest version of QTMS radar, and not to classical radar), a substantial improvement in the probability of detection through far fewer channels. Finally, we simulate signal transmission to target in QTMS radar and achieve a huge increase in QTMS radar range, from half a meter in the conventional JPA to 482 m in the current study.