This paper addresses the estimation of fractional delay and Doppler shifts in multipath channels that cause doubly selective fading-an essential task for integrated sensing and communication (ISAC) systems in high-mobility environments. Orthogonal Time Frequency Space (OTFS) modulation enables simple and robust channel compensation under such conditions. However, fractional delay and Doppler components introduce inter-path interference, degrading estimation accuracy. We propose a two-stage estimation method based on Prony's technique using OTFS pilot signals with M subchannels and N pilot repetitions. In the first stage, Doppler frequencies are estimated by jointly solving M coupled Prony equations, exploiting the periodicity of the pilot signal. In the second stage, delays are estimated by applying the discrete Fourier transform (DFT) and Prony's method to each Doppler component obtained in the first stage. The proposed method can accurately estimate up to N-1 delay-Doppler parameters under noiseless conditions. In noisy environments, conventional information criteria such as AIC and BIC yield suboptimal performance; thus, a heuristic model order selection is adopted. Numerical simulations confirm that the proposed method achieves high estimation accuracy, highlighting its potential for future ISAC frameworks.