Fault diagnosis is extremely important to the safe operation of Lithium-ion batteries. To avoid severe safety issues (e.g., thermal runaway), initial faults should be timely detected and resolved. In this paper, we consider parallel-connected battery cells with only one voltage and one current sensor. The lack of independent current sensors makes it difficult to detect individual cell degradation. To this end, based on the high-frequency response of the battery, a simplified fault detection-oriented model is derived and validated by a physics-informed battery model. The resistance of the battery string, which is significantly influenced by the faulty cell, is estimated and used as the health indicator. The statistical resistance distribution of battery strings is first analyzed considering the distribution of fresh and aged cells. A fault diagnosis algorithm is proposed and the thresholds (i.e., 2 standard deviation interval) are obtained through statistical analysis. Monte Carlo simulation results show that the proposed fault diagnosis algorithm can balance false alarms and missed detections well. In addition, it is verified that the proposed algorithm is robust to the uniform parameter changes of individual battery cells.