I think my father was, I believe, a little drunk at the time.
Literal
Father [topic-は], thinking, that-time a-little was-drunk [explanatory-のだ] [quotative-と] think.
思うに is a parenthetical insertion meaning 'I think / in my opinion.' The sentence has two layers of thought: 思うに (I believe) and ~と思う (I think). ~のだと思う wraps the whole as an explanatory inference. The hedging is very Japanese — tentative and layered.