Preterm infants have significantly higher rates of functional limitations and are at risk for delays in cognitive, motor, and other skills. Through high-risk infant follow-up (HRIF) programs and other early detection methods, these delays and complications can be reduced. However, these interventions require substantial resources and can create extra burden on families. In this paper, we present the results of a qualitative design study to understand the needs of these families and their professional caregivers. These results reveal diverse challenges facing parents and clinicians attempting to document, understand, and share infant health data that must be balanced in any pervasive health solution. In particular, parents described struggling with when to collect data, while clinicians reported being more concerned about what and how parents track infant health. Based on these results, we present five design implications for applications for preterm infant care, which we incorporated into the design of Estrellita, a mobile health informatics tool to support caregivers of preterm infants.