It is possible with a good reason not to tell your parents that at 16 a therapist might be able to talk to you without involving them. Usually, though, a parent would have to sign consent for your treatment. Good therapists do what they can to keep your confidentiality, but will encourage you to tell your parents if there is something they should know. They will also encourage your parents to respect your confidentiality. Therapists have a responsibility to keep people safe, so they have to report if they feel in their best judgement you are in danger (or someone else is) or if they know about abuse. They will keep your best interests in mind, and will try to help you to do what is best for you. Therapists do know that suicidal thoughts are a symptom that needs to be treated, and not everyone with those thoughts is in danger of acting on them. Part of their job is to assess whether you are actually at risk for suicide vs having some thoughts that you aren't going to act on.

You might be able to get a referral from your school counselor. Another place to ask might be a planned parenthood clinic. They often have counseling available, and it doesn't have to be about reproduction, and they are very used to keeping confidentiality there. If you call and ask, they may be able to help, and would certainly be able to help you find other options. You might also call the Division of Child and Family Services (or whatever they go by in your state) and ask them for referral information. Community mental health centers are another possibility, although many of them can only provide services to people with Medicaid these days.