Protein deficiency has been demonstrated to increase healing time. This has been noted in alcoholics who notoriously imbibe their nutrition by the bottle. Caloric deficiency in a mild state (dieting) has not been demonstrated to reduce healing time.
Pars and facet fractures tend not to heal. The best chance of healing is to avoid any significant motions of the lumbar spine. A flat back brace is what I use for patients with pars fractures but these braces won’t prevent deleterious motions, just remind patients to avoid the motions.