Radial velocity (RVel) measurements of transiting multiplanet systems allow us to understand the densities and compositions of planets unlike those in the solar system. Kepler-102, which consists of five tightly packed transiting planets, is a particularly interesting system since it includes a super-Earth (Kepler-102d) and a sub-Neptune-sized planet (Kepler-102e) for which masses can be measured using RVs. Previous work found a high density for Kepler-102d, suggesting a composition similar to that of Mercury, while Kepler-102e was found to have a density typical of sub-Neptune size planets; however, Kepler-102 is an active star, which can interfere with RVel mass measurements. To better measure the mass of these two planets, we obtained 111 new RVels using Keck/HIRES and Telescopio Nazionale Galileo/HARPS-N and modeled Kepler-102's activity using quasiperiodic Gaussian process regression. For Kepler-102d, we report a mass upper limit Md<5.3M{Earth} (95% confidence), a best-fit mass Md=2.5{+/-}1.4M{Earth}, and a density {rho}d=5.6{+/-}3.2g/cm^3^, which is consistent with a rocky composition similar in density to the Earth. For Kepler-102e we report a mass Me=4.7{+/-}1.7M{Earth} and a density {rho}e=1.8{+/-}0.7g/cm^3^. These measurements suggest that Kepler-102e has a rocky core with a thick gaseous envelope comprising 2%-4% of the planet mass and 16%-50% of its radius. Our study is yet another demonstration that accounting for stellar activity in stars with clear rotation signals can yield more accurate planet masses, enabling a more realistic interpretation of planet interiors.