I agree, the best way to do this is to figure out WHY your Dad is saying "NO" and then convince him the prohibition isn't necessary.
Do you have a driver's license? A car is just as much a responsibilty as a gun is, both wield the power of life and death. A driver's license and a clean driving record show responsibility. Got good grades? Never been in dutch with the Law? Don't get held after at school? Keep up on your chores all right? Working some kind of job? Known for not taking stupid "kid" risks? Always in before curfew? Your whereabouts never in question? run with a respectable crowd? All of these will weigh in your favor. Hopefully, as I did, by the time your 18 you have 3 years or more of this kind of track record. That's what I meant by "good boy" credits. If you can't answer YES to just about all of these, then you have bigger fish to fry than messing with firearms anyway. Buckle down and start building a reputation, then check back in a year or so and see where it's gotten you.
If you get the FOID card, at least give Pops a chance to come around. Take a safety class. Go shooting regularly with Dad or with someone Dad respects (i.e. Wally Cleaver, not Eddie Hascal). After a few months of this (yeah, I know, patience is a virtue but it isn't a lot of fun...) revisit the issue with Dad. If the answer is no, then never speak of it again. If you do decide to go behind your Dad's back and get one anyway, then you must be EXTRA careful that you never do anything that would get you in trouble with it. You must never let Dad find out, either. One can have the opinion that Dad is getting what he wants (you not having a careless incident with a firearm) but just not in the way he stipulated. Admit to yourself that this is technically "wrong" but it is the only way to get a fair shake for both of you.
Keep in mind that if Dad or Mom or even Grandpa (who lives in the spare bedroom) have a felony (even if no time was served, plea bargain) from decades ago, or a domestic violence conviction (even if the charges were bogus) you could be subjecting them to VERY REAL RISK of jail time if a firearm is found in the house with them. You think you know everything about your parent's past? Most of their lives were lived before you were born, likely. There may be some skeletons in the closet they don't want you or anyone else to know about. Think carefully about what you do, it's not just yourself you have to think about. Also consider that you may have others in the house who are not as trustworthy as you - for instance a little brother, or a friend who comes over occasionally, who they do not trust as much as they trust you. If something bad happens (little bro or your careless friend find your gun, pop a round in the chamber and proceed to have a negligent discharge) your folks will be held accountable.