駐車場の駐車スペースが空いていたので、待たずにすみましたって英語でなんて言うの？

All of these options are correct and valid when explaining that there was no parking available.
You may also choose to explain at which parking lot you were waiting/looking for a space at, e.g. "The parking lot at the supermarket was full, so I decided not to wait"
You may also decide to expand the sentence and say who or what you were waiting for, e.g. “The parking lot at the cinema was full, so I decided not to wait for my friends”

"I was able to get a parking spot without waiting, today."
Get - you were able to obtain something
Without waiting - There was no delay, it was quick.

You could also say,
"I was able to get a parking spot without circling the parking lot, today."
Circling - meaning to drive around the parking lot until there was an opening for you to park.

When talking about parking your car at a supermarket, and the parking lot is usually full and you have to wait a little to get a spot - but today there are many free parking spaces and it is not necessary to wait, then just use the abnove exampled sentence! The first is in the present tense, the second in the past tense.
In the above sentences, the context - ie a carpark and you being the driver of a car, is already known to the speaker and listener.

When explaining to someone that you did not have to wait for a parking spot at the supermarket because it was empty, you can say something like "The supermarket was empty and so i got a parking spot immediately without waiting." or "I didn't have to wait for a parking spot because the parking lot was empty today.". Both sentences mean the same thing and are polite ways of saying this.

An opening you park your car in in a car park is also called a 'space' or 'car parking space'
so to expain you didn't have to wait you would say 'I was able to get a parking soace without waiting today'

Saying that getting a 'park' was super easy does not give the exact reason why parking was easy however; it does let the person listening know that you didn't have to wait.

People who speak English assume allot of details. Therefore when you say that there was no reason to 'wait around,' it lets the person who is listening know that there was no difficulty in finding a parking spot.