Parent-child story time is an important ritual of contemporary parenting. Recently, robots with artificial intelligence (AI) have become common. Parental acceptance of children's storytelling robots, however, has received scant attention. To address this, we conducted a qualitative study with 18 parents using the research technique design fiction. Overall, parents held mixed, though generally positive, attitudes toward children's storytelling robots. In their estimation, these robots would outperform screen-based technologies for children's story time. However, the robots' potential to adapt and to express emotion caused some parents to feel ambivalent about the robots, which might hinder their adoption. We found three predictors of parental acceptance of these robots: context of use, perceived agency, and perceived intelligence. Parents' speculation revealed an uncanny valley of AI: a nonlinear relation between the human likeness of the artificial agent's mind and affinity for the agent. Finally, we consider the implications of children's storytelling robots, including how they could enhance equity in children's access to education, and propose directions for research on their design to benefit family well-being.