Tattoo sleeves on women have gained in popularity and therefore aren't quite as shocking, but unless your boyfriend comes from a family of tattooed people you might add a little shock value to the family functions, not that there's anything wrong with that.
How do you plan on dressing otherwise? Tattoos paired with scanty clothing are not going to make the best impression or look appropriate so choose your dress in a more classic fashion and you can help compensate for the bold appearance of your tattoos. Tailored and classic styles are your best option. Avoid anything printed or too colorful. Simple solid colors will always look the most sophisticated. Consider a black sheath dress and a cardigan for easy elegance with a but of edge.
Another consideration is how do your tattoos compare to your style otherwise? Many people have tattoos that work well with their overall fashion style and that helps to make a more complete statement rather than just a few scattered random tattoos on your arm. If you have pin-up tattoos then by all means dress the part a little bit. You're more likely to make a good impression when you have good style regardless of how many tattoos or where they are placed.
Even then, you'll likely still make the impression of a fun, maybe a little wild, individual who is daring and freespirited. Hopefully you can relax and just have a good time and not worry so much about what his parents think. You're more likely to make your best impression by just being yourself. You are who you are and they are either going to take to you or not. If you feel most attractive bearing your body art then you likely already wear your tattoos well. And you should, especially if you put proper thought into the designs in the first place. With confidence is how you should carry yourself. Be outgoing and friendly and just be you. Your tattoos do not define you, they are a part of you.
If you plan on hiding your tattoos you can opt for shawls and wraps on even bracelet-sleeve dresses which all work well into the holiday season.
Revealing tattoos at other social functions such as holiday workplace parties and corporate events is subject to any workplace agreements regarding body modifications and tattoos. Use your best judgment and always have a few black dresses on hand with full sleeves and a knee length so that you can cover up on the fly and still look elegant, no matter the occasion.