If the person is under the age of 18 he or she may not get a tattoo or pierced earring even with with parents consent or a parent present.

However if a person is refused a tattoo over the age of 18 the owner or the person must give a reason without being
discriminatory such as color race religion.

They can also refuse you if you have the following diseases such as

HIV
TB
Sexual disease

you are required to let that tattoo parler know under New York state law

But if your a convicted felon they cannot use that to not give you a tattoo unless you are a registered Sex offender and you must let them know.because you are changing your identity and you must notify police agency of any new tattoo and some of these people do not report them.