Good of the authors to explicitly acknowledge that there is no causal link indicated here.
Fun Hypothesis: There exists an underlying latent variable that determines both of these. It takes the form of a negative affect activation function; ie. at some level of pain/annoyance we start to consider the source a problem and look for solutions. Adding this latent variable to the analyses would mediate the relationship between acetaminophen use and the child's behavioral 'issues'.
The authors do address this a bit with:
"They looked for a link between a child’s behavioral problems at age 7 and his or her mother’s post-natal acetaminophen use, and found none."
I would not suppose that this activation function is linear which may confound the authors attempt to control for this.