hello Julie,
I am a woman who first walked the camino at 59 and I feel that albergues(also called refugios) are among the safest places you could stay. After all, everyone has to be in and in bed by around 10 o'clock. You are sleeping in a room with 6,8, 24 or more other people which has to be a lot safer than sleeping alone. Your parents can feel reassured that you will be surrounded by people who are willing to help you.
I've never walked the Portugues Route, which I guess is what you're planning to do since you say you're walking from Porto. In July of 2011 over 3000 people walked the Portugues Route so you can see you will have lots of company if you wish. Of course you will have to take the usual precautions that women everywhere have to take any time but the camino feels safer to me than some places in my home country (Canada). You won't ever have to walk alone if you don't want to. There is some discussion about safety for females on this forum under the Camino Portugues.
I don't know how it is in Portugal but in Spain there are a variety of places you can stay apart from albergues. They are: pension(usually a shared bathroom down the hall), casa rural(like a B&B- private room and bathroom), hostal( about like a 2 star hotel - simple, usually clean, private bath but no frills), hotels
Good luck with your planning and buen camino.