"In his letter to us he stated that he was suffering from acute depression and he felt it was his only way out," the parents of the The Bachelorette contestant tell PEOPLE in a statement. "While publicly outgoing he was an extremely private person who internalized everything."
Confirming that Hug, 35, committed suicide earlier in the week in a remote area, Bertrand and Denise Hug say, "We feel it is necessary at this time to address the media as we are devastated by some of the comments that are coming out."
Julien died from a self-inflicted gunshot wound, a family spokeswoman says. In addition to leaving a note for his parents, he left letters for his coworkers and his girlfriend.
"Our son meant the world to us and this came as a complete shock," his parents say. "We are beyond devastated and completely distraught. … What we saw was his thirst for life with fast cars, motorcycles and his love of skydiving."
Julien's body was discovered this week in a remote part of the San Bernardino National Forest in Southern California after he was reported missing on Monday.
"What we see now is the immense amount of love that everyone has for him," his parents say. "Anyone close to Julien is receiving an extreme outpouring of concern via text, email and messages. This is an honored tribute to what an amazing person he was and we thank everyone for their love, prayers and continued support."
"To us Julien was as beautiful inside as he was outside and we wish to have him remembered in this way, a way he deserves and would appreciate, not by false comments from people outside the family. No one he has touched will ever forget him, most of all us."