The research aimed to evaluate the efficacy of the NeuroAssist, a parallel robotic system comprised of three robotic modules equipped with human–robot interaction capabilities, an internal sensor system for torque monitoring, and an external sensor system for real-time patient monitoring for the motor rehabilitation of the shoulder, elbow, and wrist. The study enrolled 10 consecutive patients with right upper limb paresis caused by stroke, traumatic spinal cord disease, or multiple sclerosis admitted to the Neurology I Department of Cluj-Napoca Emergency County Hospital. The patients were evaluated clinically and electrophysiologically before (T1) and after the intervention (T2). The intervention consisted of five consecutive daily sessions of 30–45 minutes each of 30 passive repetitive movements performed with the robot. There were significant differences (Wilcoxon signed-rank test) between baseline and end-point clinical parameters, specifically for the Barthel Index (53.00 ± 37.72 vs. 60.50 ± 36.39, p = 0.016) and Activities of Daily Living Index (4.70 ± 3.43 vs. 5.50 ± 3.80, p = 0.038). The goniometric parameters improved: shoulder flexion (70.00 ± 56.61 vs. 80.00 ± 63.59, p = 0.026); wrist flexion/extension (34.00 ± 28.75 vs. 42.50 ± 33.7, p = 0.042)/(30.00 ± 22.97 vs. 41.00 ± 30.62, p = 0.042); ulnar deviation (23.50 ± 19.44 vs. 33.50 ± 24.15, p = 0.027); and radial deviation (17.50 ± 18.14 vs. 27.00 ± 24.85, p = 0.027). There was a difference in muscle activation of the extensor digitorum communis muscle (1.00 ± 0.94 vs. 1.40 ± 1.17, p = 0.046). The optimized and dependable NeuroAssist Robotic System improved shoulder and wrist range of motion and functional scores, regardless of the cause of the motor deficit. However, further investigations are necessary to establish its definite role in motor recovery.