If one is depressed can they listen to music during their year of aveilus for a parent?

Thank you for your question, Yes you could. Try to make sure that people that don't know that you have permission to hear music don't see that you hear music during the aveilus (Harav ovadia Yosef Maayan omer part 4, siman 219). If you could hear calm music that's preferable since it might be permitted in any case to hear calm backround type of music (See Dvar Yehoshua part 3, 63). All the best!