It's not obligatory. You could easily say "Ele au inceput ziua razand".
As a native, it feels like putting "si-au" instead of "au" makes the action seem centered on them and how their day in particular was going.
If it was "Ele au inceput ziua razand", it would translate to "They started THE day laughing".
Also, Duolingo likes to put the pronoun when in actual speech it is only used when it's not clear who we're talking about. If the previous sentence is also about them, you would normally say "Si-au inceput ziua razand", without the pronoun.