To clarify the unclear antecedents in the passage, the most effective revision is:
"Enrico had to work from home and take care of his two-year-old son. They walked over to the park, and while there, Enrico planned to make a call to his co-worker Paul. Helping his son on the playground slide, Enrico talked to Paul using a hands-free headset."
This revision ensures that the antecedent "he" clearly refers to Enrico when describing the call to Paul.