Anesthesia in childbirth? Please tell me you're talking about epidurals, and not full anesthesia, which is only given if there are serious problems with the childbirth and major surgery needs to happen. We don't live in the 19th century any more. Further, it wouldn't deter those women who don't plan on having children. We do exist, you know. Also, the assertion that tattoo ink causes a bad interaction with painkillers (but only during childbirth!) sounds extremely suspect to me.
It's true. If you have ink in the "right" spot on your lower back, that's fresh, you can't have an epidural. I know this because my lower back tattoo was just high enough to not be in the wrong place for epidural placement, and because it was over 4 years old. So there is some fact to what they were saying.
ETA: The doctor told me that I could have one, but proceeded to explain about new ink and wrong placement interfering with epidural placement.