Learn Italian: Quanto ci metti a parcheggiare la macchina? - How long does it take you to park the car?

Metterci is a personal verb, meaning it focuses on how much time a specific person takes. Because the sentence asks how long you take to park, we conjugate it for tu (ci metti). If we used the impersonal ci vuole, the question would mean "How long does it take in general to park the car?"

What is the purpose of the ci in ci metti? Does it mean "us" or "there"?

No, in this context ci doesn't have its usual meaning of "us" or "there". Instead, it's a permanent part of the verb metterci. Adding ci changes the regular verb mettere (to put) into a special verb meaning "to take an amount of time".

Why is there an a between metti and parcheggiare?

When you follow metterci with an action (an infinitive verb like parcheggiare), you must link them with the preposition a. You will always use metterci a before another verb to express what action is taking time.

Can I say Quanto tempo instead of just Quanto?