Damage identification methods for engineering structures based on vibration parameters have the advantages of easy detection and high precision; however, structural strain information is more sensitive to structural damage than displacement information. Traditional resistance strain sensors have low accuracy and poor stability when measuring structural strains. Therefore, this paper uses a highly sensitive polyvinylidene fluoride dynamic strain sensor to identify structural damage in a thin plate. The polyvinylidene fluoride sensor is used to obtain structural strain response information, and structural modal parameters are identified using operational modal identification methods based on the natural excitation technique and the eigensystem realization algorithm. This paper uses a damage index based on mode shape and flexibility. A new damage index based on the LU decomposition of the flexibility matrix is used to identify the damage of the thin plate structure. The effectiveness of the modal identification methods and the new damage index is validated via an elastic thin plate experiment. The results show that the modal identification method and the new damage index proposed in this paper can identify damage in a thin plate structure. Sensor comparison experiments also show that compared with a resistance strain sensor, the polyvinylidene fluoride sensor has higher damage sensitivity, better damage recognition and the ability to recognize farther from the sensor.