We report the discovery of a super-Earth and a sub-Neptune transiting the star HD 15337 (TOI-402, TIC 120896927), a bright (V = 9) K1 dwarf observed by the Transiting Exoplanet Survey Satellite (TESS) in Sectors 3 and 4. We combine the TESS photometry with archival High Accuracy Radial velocity Planet Searcher spectra to confirm the planetary nature of the transit signals and derive the masses of the two transiting planets. With an orbital period of 4.8 days, a mass of {7.51}-1.01+1.09 {M}\oplus and a radius of 1.64 ± 0.06 R ⊕, HD 15337 b joins the growing group of short-period super-Earths known to have a rocky terrestrial composition. The sub-Neptune HD 15337 c has an orbital period of 17.2 days, a mass of {8.11}-1.69+1.82 {{{M}}}\oplus , and a radius of 2.39 ± 0.12 R ⊕, suggesting that the planet might be surrounded by a thick atmospheric envelope. The two planets have similar masses and lie on opposite sides of the radius gap, and are thus an excellent testbed for planet formation and evolution theories. Assuming that HD 15337 c hosts a hydrogen-dominated envelope, we employ a recently developed planet atmospheric evolution algorithm in a Bayesian framework to estimate the history of the high-energy (extreme ultraviolet and X-ray) emission of the host star. We find that at an age of 150 Myr, the star possessed on average between 3.7 and 127 times the high-energy luminosity of the current Sun.