We investigate the long-term persistence of solar energetic particle (SEP) time series by means of four different methods: Hurst rescaled range R/S analysis, detrended fluctuation analysis, centered moving average analysis, and the fluctuation of natural time under the time reversal method. For these analyses, we use data sets from the Integrated Science Investigation of the Sun instrument suite on board NASA's Parker Solar Probe. Background systematic noise is modeled using cross-correlation analysis between different SEP energy channels and subtracted from the original data. The use of these four methods for deriving the time-series persistence allows us to (i) differentiate between quiet- and active-Sun periods based on the values of the corresponding self-similarity exponents alone; (ii) identify the onset of an ongoing activity well before it reaches its maximum SEP flux; (iii) reveal an interesting fine structure when activity is observed; and (iv) provide, for the first time, an estimate of the maximum SEP flux of a future storm based on the entropy change of natural time under time reversal.