I noticed in the links that Eoin Joyce provided, the Bible references never mention a tattoo in them. It did give reference to obeying your parents (Colossians 3:20), which I can see that, but no mention of not getting a tattoo once an adult.
1 Peter 3:3&4, these are speaking about a wife and how she should be submissive to their husband throughout the first 6 verses. In fact it is specifically addressed in verse 5. Again, nothing about tattoo's. This applies also to 1 Timothy 2:9 as it is stated again about submissiveness and reinforced at verse 11.
Romans 12:1, this is the only verse that can actually pertain to the subject, however it still does not mention tattoo's. Romans 12:1&2 speak to us about how we are to live in these bodies as a living sacrifice, acceptable to God. We are to live for God, this would apply to smoking, drinking in excess, fornication, slandering and even over eating as these have been mentioned in other scriptures, but does it really apply to tattoo's? I think not.
There is only one mention of tattoo's in the Bible. Leviticus 19:28. However, the WT accepts that this is not binding to anyone but the Jews.
https://www.jw.org/en/bible-teachings/questions/bible-say-about-tattoos/#?insight[search_id]=aa3524b3-b921-40f1-b8fd-549dc406c9c3&insight[search_result_index]=1
" While the Law given to Israel is not binding on Christians, the principle underpinning this law is worth serious consideration. "
Worth consideration does not equate to absolute NO.