Existential fear is fear arising from a growing and undeniable recognition that death is inescapable. It is more than a logical realization. It is the deep, gut level personal truth that feels the fact that almost everyone who has ever existed is now dead and that the same fate awaits me and everyone I know. When teenagers have this experience it can be very frightening and parents can be very helpful in assisting their teenagers to cope with it.
Young teenagers in particular can be very distressed by the visceral realization that they will one day die and may start speaking a lot about death, macabre subjects, and generally exploring the darker side of life. The important thing for parents to keep in mind is that this experience is normal in teenagers. It is a developmental stage that encourages the development of the teenager's individual way of understanding the nature of life and creating a personal, meaningful philosophy about it. This can be a rocky phase for teenagers but parents should allow their teenagers to explore this and be open to talking about it. Some parents feel that conversations with their teenagers about the darker side of existence is harmful to the teenagers but the opposite tends to be true. If death is something that cannot be openly discussed you may create a sense of taboo in your teenager that can interfere with their attempts to understand and cope.
Expressions of love are very important at this time. Feelings of being loved and important to parents takes some of the edge off and allows the teenager to explore their fears in a more productive way. It can be helpful to speak to your teenager about the ways in which you as an adult cope with the certainty of eventual death. This can take the shape of religious or spiritual beliefs or whatever philosophy or strategy is helpful to you. Say these things as an example of how one person deals with death but don't get pushy or preachy. Do not pressure them to cope in the same way you do, especially if you feel resistence. You should say that it is everyone's personal quest to answer the question of how to cope with the knowledge of their own deaths and that you as a parent will be there to support their efforts to do this with love and acceptance.
A certain amount of brooding and time alone thinking is normal when teenagers face this challenge. Some light and loving checking in with them might be all that is required here. However, if you notice that grades are dropping, that they are loosing interest in things, that they feel stuck, have stopped socializing or talk about killing themselves it is a sign that your teenager needs extra help. Psychotherapy can be very helpful to both teenagers and parents in this situation and there is no responsible reason to postpone starting.