Fire brigade chaplain Greg Griego, 51, his wife Sara, 40, and children Zephania, nine, Jael, five, and Angelina, two, died in the shootings, which took place on Saturday night in a rural area south-west of central Albuquerque.
Mr Griego's 15-year-old son has been charged with two counts of murder and three counts of child abuse resulting in death.
Authorities would not reveal any details about conversations the teenager had with investigators.
However, it was revealed each victim sustained more than one gunshot wound and several guns were found at the home, including a semi-automatic military-style rifle.
Fire Chief James Breen described Mr Griego as a "dedicated professional" known for his work with firefighters and as a volunteer chaplain at the county jail.
The killings come just three days after President Barack Obama presented a sweeping package to tighten gun laws in the US following last month's school shooting in Connecticut, which left 20 children and six adults dead.