The revised passage that corrects any unclear antecedents 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 clarifies that Enrico is talking to Paul, not his son, while helping his son on the slide.