This is a big no-no. I own a tattoo studio and one of the questions on the health questionnaire you fill in before getting anything (tattoo or piercing) is whether you could be pregnant, and if the answer is yes, the form goes in the bin and we tell you to come back after the baby is born! (Ideally not within the first six months and not while you're breastfeeding)
Even though a person may not find it painful, tattooing still puts a huge stress on the body, it is a big skin wound basically and then needs to heal. The main reason against it is the risk of introducing infection into your body, especially as your immune system is reduced due to pregnancy (same goes for piercing, even more so as they take a lot longer to heal than a tattoo). You also may not react the same way you usually would to the ink, so colours may not look the way they should or you may be sensitive to it even if you're not usually, which can cause ink to reject and your tattoo to look patchy. Your body is already under a huge amount of pressure, growing a new human being, so you want to avoid putting any further stress on it if you can!
I did actually get both tattooed and pierced in the very early stages of this pregnancy, before I knew. Three months later, the tattoo has only just healed (they usually take 2-3 weeks) and the piercings show no signs of settling down anytime soon!
I'm in Australia tho so maybe it's different else where
