The Gaia Data Release 3 (DR3) contains reflectance spectra at visible wavelengths for 60,518 asteroids over the range between 374-1034 nm, representing a large sample that is well suited to studies of asteroid families. We wanted to assess the potential of Gaia spectra in identifying asteroid family members. Here, we focus on two L-type families, namely Tirela/Klumpkea and Watsonia. These families are known for their connection to Barbarian asteroids, which are potentially abundant in calcium-aluminum rich inclusions (CAIs). The method we developed to establish family memberships is based (1) on a color taxonomy specifically built on Gaia data and (2) on the similarity of spectra of candidate members with the template spectrum of a specific family. Our work demonstrates the advantage of combining the classical hierarchical clustering method (HCM) approach to spectral properties obtained by Gaia for the study of asteroid families. Future data releases are expected to further expand the capabilities in this domain. The memberships for the Tirela/Klumpkea and Watsonia families are reported here. The columns report, from left to right: the identifier of the asteroid, the absolute magnitude, the proper elements (semi-major axis, eccentricity and sine of the inclination, taken from AFP, Novaković et al., 2022), NEOWISE albedo (Masiero et al., 2011) and spectral type from our color taxonomy. For the objects that are not directly classified into the S and L classes, their most probable spectral type is also reported.