'Lyric Angel' should be the name of a website that you can look up to find out the words to your favorite musician's latest tracks. It could also be a nickname given to someone who's exceptionally good with the written word. When you give that name to someone who's just been born, you're putting a lot of pressure on them when it comes to their future career choices.
The combination of 'lyric' and 'angel' wasn't necessarily a deliberate choice - his father is former boyband singer Ashley Parker Angel who was once in O-Town - but flipping 'Parker' and 'Angel' around suggests there was some intent there on some level between himself and his then-partner, model and actress Tiffany Lynn. Lyric is an aspiring nineteen-year-old actor who has worked on shows like 'American Horror Story.'