I'm sure you know what the 5th Amendment means to our culture and thinking without having read the whole constitution. In fact it's an amendment, the actual meat of the constitution is largely unknown to most of the population and mostly irrelevant to most of our culture. (If you are not American, sorry)

So read the most relevant OT books and get on with it! Genesis, Exodus, Duteronomoy, Ruth, a little of the major prophets, Job, Proverbs 8, Psalm 22,23,91 I think? I'm probably missing a lot but seriously it's not needed if you want to understand NT context. I mean the rabbinical culture that Jesus was dealing with in the Gospels was based on a side-conversation the Jews were having about the Torah, not even about the Torah itself. So you'd have to read the Mishnah as well if you really wanted the full context.

I think it's safe to say that analyzing any historical work requires a bit of back referencing. I believe I watched Spaceballs before watching Star Wars, and I think I enjoyed Star Wars more because of it. Can't say for sure, it was a long time ago.