A tunable narrow linewidth fiber ring laser based on parity-time (PT) symmetric double rings and variable optical attenuator (VOA) is demonstrated. The laser is composed of a fiber loop mirror (FLM), which creates two coupled rings: the reflective ring and the transmissive ring. The loss of the transmissive ring can be controlled by tuning the VOA, which balances with the gain of the reflective ring. When the PT symmetry is broken, a single-longitudinal-mode (SLM) laser is realized. The laser can be tuned by an optical tunable filter (OTF), and the tuning range is 38 nm, spanning from 1525 nm to 1563 nm. In the whole tuning process, the linewidth of the laser is at the sub-kHz level, and the optical signal to noise ratios (OSNR) is over 53 dB. The fluctuation of wavelength and power are less than 0.02 nm and 0.52 dB in an hour, respectively.