At 16 years old, and with a parent (your dad) willing to help look after you? You’ll potentially find most hospitals will bend over backwards to keep you out of a psych ward. The common view is that they are not a therapeutic place for young adults, and may very easily do more harm than good.
So, even where there are some suicidal thoughts, as distressing as it is, it may be the case that they don’t consider hospitalisation to be the best option for you. Especially if dad is prepared to help keep you safe at home while you work with your T to heal, and overcome your suicidal thoughts.
Hospitalisation for suicidality is likely to simply be a holding-place, where you lose your privacy and freedom to come and go, and are supervised, until it is considered you are no longer an acute risk. So if dad has told your therapist theat he’s prepared to supervise you, many hospitals are going to be reluctant to admit you at 16.
If you can, perhaps have another conversation with your T, and ask them to be a bit more specific about why you haven’t been admitted, and perhaps review your concerns with them. If you think you are currently not able to keep yourself safe, it is really important that you’re communicating this with someone like your dad or your T, so that they can provide the appropriate level of care.