Hyades Member K2-136c: The Smallest Planet in an Open Cluster with a Precisely Measured Mass

K2-136 is a late-K dwarf (0.742 ± 0.039 M _⊙) in the Hyades open cluster with three known, transiting planets and an age of 650 ± 70 Myr. Analyzing K2 photometry, we found that planets K2-136b, c, and d have periods of 8.0, 17.3, and 25.6 days and radii of 1.014 ± 0.050 R _⊕, 3.00 ± 0.13 R _⊕, and 1.565 ± 0.077 R _⊕, respectively. We collected 93 radial velocity (RV) measurements with the High-Accuracy Radial-velocity Planet Searcher for the Northern hemisphere (HARPS-N) spectrograph (Telescopio Nazionale Galileo) and 22 RVs with the Echelle SPectrograph for Rocky Exoplanets and Stable Spectroscopic Observations (ESPRESSO) spectrograph (Very Large Telescope). Analyzing HARPS-N and ESPRESSO data jointly, we found that K2-136c induced a semi-amplitude of 5.49 ± 0.53 m s^-1, corresponding to a mass of 18.1 ± 1.9 M _⊕. We also placed 95% upper mass limits on K2-136b and d of 4.3 and 3.0 M _⊕, respectively. Further, we analyzed Hubble Space Telescope and XMM-Newton observations to establish the planetary high-energy environment and investigate possible atmospheric loss. K2-136c is now the smallest planet to have a measured mass in an open cluster and one of the youngest planets ever with a mass measurement. K2-136c has ~75% the radius of Neptune but is similar in mass, yielding a density of ${3.69}_{-0.56}^{+0.67}$ g cm^-3 (~2-3 times denser than Neptune). Mass estimates for K2-136b (and possibly d) may be feasible with more RV observations, and insights into all three planets' atmospheres through transmission spectroscopy would be challenging but potentially fruitful. This research and future mass measurements of young planets are critical for investigating the compositions and characteristics of small exoplanets at very early stages of their lives and providing insights into how exoplanets evolve with time.