UK Citizen by Descent Without a UK Passport

[chimpledfoothill]

Unfortunately Craig you are not a British citizen, your parents may be British but you, presumably, were born in the US, have an American passport and therefore are an American citizen.
You state that you want to know how to prove your UK citizenship upon arrival in Scotland. You can't because you do not have UK citizenship. Informing immigration officers that your parents are British is irrelevant, it will have no bearing on the decision to grant you entry. Your only option is to enter as a tourist and with all the restrictions that that entails. In order to remain, work, live as a British citizen you will have to apply for citizenship. I have no experience of the process so I cannot offer any advice in that respect.

Apply for the UK Passport first. It will in the end be much less rather more hassle to do it this way. You do need to submit your American passport as part of the application process, as well as appropriate birth certificates etc. These days there are a number of times when you need to prove your right to live in the UK (eg to rent property or work).