The curved beak along with the fleshy, agile tongue are some of the telltale features of parrots. It is a very powerful but precise multi tool used for heavy duty destruction and delicate fine motor tasks alike: from cracking nuts and chewing through wood to preening, feeding delicate babies and de-hulling tiny seeds with great precision. It is also used as defense, and in some cases to inflict damage by nipping or biting.
Parrots open their beak in many different contexts:
When reaching for something to eat or explore
When climbing or stepping on to a new surface to help keep their balance
When defending themselves
Before they bite
When panting, for example when very warm and/or after rigorous exersize
Generally, a parrot that is reaching for something to eat or play with will open it's beak just enough to grab on to that item, and generally extend it's tongue to investigate. A parrot in a defensive posture and/or preparing for a possible bite will often open it's beak pretty wide, although in some contexts even a slightly open beak can be a sign that the bird is uncomfortable. They will usually also retract the tongue to keep the sensitive body part out of harms way when using the beak defensively or to inflict damage. A very warm parrot will usually be breathing rapidly and you can often see the tongue moving up and down.
Left: This macaw is reaching for a twig with it's beak. The tongue is clearly visible. Right: This macaw is in a defensive posture. It's tongue is tucked away and not visible.
1.5 BEAK AND TONGUE
Feaking is a term for when birds rub the side of their beak on a perch or other object. They often do this after eating, during or after preening, or sometimes as a displacement activity.
Most parrots will make a crackling noise by grinding their upper and lower beak against each other, most often in conjunction with feaking, when resting or when settling down to sleep. This is often referred to as beak grinding.
This is when a parrot repeatedly opens and closes the beak resulting in a clicking or "clacking" sound. Beak clicking can be slow or very fast, and is observed especially in cockatoos, including cockatiels. They might click their beaks in a variety of contexts, such as when cuddling and asking to be scratched, or when they are very aroused.