There is no state of New Orleans
The only state that is not is Louisiana. They are divided into Parishes instead of counties.
Louisiana uses parishes instead of counties.
You mean "What US state has parishes not counties"; and the answer is Louisiana.
Louisiana
There are 48 states with counties and 2 states with parishes. States like Louisiana and Alaska have parishes instead of counties.
parishes
Louisiana
Initially the state of Louisiana was divided into counties, but these were poorly organized and it was found the parishes that had been established by the France and Spain, both Catholic countries worked better as a means of division. Louisiana's parishes are based on a method of ecclesiastical division.
special districts
It is the same, just a different name. Some are also boroughs.
Parish [instead of counties, Louisiana has parishes].
Generally speaking, a parish is a group of members of a specific church. In the USA all its States have churches and thus parishes. States also have other terms to identify themselves. They may not use the term "parish" but it's clear that members of Islamic and Jewish peoples, to give a better perspective on this, have members who gather to worship at temples and synagogues. All US States have parishes for the major religions in the USA.