For example in the sentence
There is a car park at the library with designated disabled
it is correct write
There exists a car park at the library with designated disabled
Your sentence starting "There exists ..." is grammatically correct, and it has the same meaning as the sentence starting "There is ...". However, "There exists ..." is not typically used in casual, informal speech. We associate using "there exists" with the presentation of logical proofs, as in mathematics papers.
If someone asked you "Where can I find a disabled parking space?", and you answered using the sentence starting with "There exists ...", you would sound strange; the sentence that starts with "There is .." would be normal.