HAWC J1826–128 is one of the brightest Galactic TeV γ-ray sources detected by the High Altitude Water Cherenkov (HAWC) observatory, with photon energies extending up to nearly ~100 TeV. This HAWC source spatially coincides with the H.E.S.S. TeV source HESS J1826–130 and the "Eel" pulsar wind nebula (PWN), which is associated with the GeV pulsar PSR J1826–1256. In the X-ray band, Chandra and XMM-Newton revealed that the Eel PWN is composed of both a compact nebula (~15'') and diffuse X-ray emission (~6' × 2') extending away from the pulsar. Our NuSTAR observation detected hard X-ray emission from the compact PWN up to ~20 keV and evidence of the synchrotron burn-off effect. In addition to the spatial coincidence between HESS J1826–130 and the diffuse X-ray PWN, our multiwavelength spectral energy distribution (SED) analysis using X-ray and γ-ray data establishes a leptonic origin of the TeV emission associated with the Eel PWN. Furthermore, our evolutionary PWN SED model suggests (1) a low PWN B-field of ~1 μG, (2) a significantly younger pulsar age (t ~ 5.7 kyr) than the characteristic age (τ = 14.4 kyr), and (3) a maximum electron energy of ${E}_{\max }=2$ PeV. The low B-field, as well as the putative supersonic motion of the pulsar, may account for the asymmetric morphology of the diffuse X-ray emission. Our results suggest that the Eel PWN may be a leptonic PeVatron particle accelerator powered by the ~6 kyr old pulsar PSR J1826–1256 with a spin-down power of 3.6 × 1036 erg s–1.