Primary word stress is typologically diverse. In some languages, the metrical structure of a word predicts the location of primary stress, while in other languages it does not. This diversity is considered through the lens of Harmonic Serialism (HS), a serial constraint-based theory, and it is argued that HS must incorporate a limited degree of parallelism to capture the typology. Namely, primary-stress assignment is simultaneous with foot-building and also mobile, being (re)assessed throughout a metrical derivation. But incorporating this parallelism into HS is both possible and desirable: the positive typological consequences of HS are preserved, and the implied formal divergence between the prosodic word and the foot with respect to parallelism echoes a fundamental distinction that is visible in a wide range of extant theoretical and empirical findings.