I have to pick up my parents

Tanya, 

in this case I suspect it is mainly that the clue was to use 'falloir que', but also to ensure we don't "run away" from this much used expression - "il faut que". As it was an 'obligation', colloquial French would be to use 'il faut que', but as non-native speakers we seem often to use devoir more than native speakers would. 

With regard to the 2nd part, 'aller chercher qqn' is a fixed expression equivalent to the English 'pick someone up' - an arranged collection whereas 'chercher qqn' alone is more akin to 'look for someone'.  I particularly like the definition in French in Larousse linked below.

https://www.wordreference.com/fren/aller%20chercher

 https://www.wordreference.com/fren/chercher

 https://www.larousse.fr/dictionnaires/francais/chercher/15132

Bonjour Tanya,

 je dois chercher mes parents

vs.  il faut que je cherche mes parent

I agree with Maarten's post.

Taking the above two examples; as I see it there is no information about any action to perform the task, simply a recognition of the need or obligation.

To write "aller chercher" is a standard expression but also here we have an indication of an action of displacement (to go) involved that is actually going to find/locate the parents.

Bonne journée

Jim