How do I tell my dad I want his old DSLR?

My first question is: why do you want it in the first place?  Do you want it because you think it's cool, and you want to play with it?  Do you have a genuine interest in learning photography?

If you just want fool around with it, then just ask him.  If he says no again, then it's no big deal.  Like you said, it's quite obsolete anyway, right?  More on that later.

If you have a genuine interest in photography, then you have a much stronger reason for wanting to use your dad's camera.  If you show your father that you are taking this hobby seriously, and that the phone isn't good enough for what you need, then it's likely that your dad will say yes providing that over the past 4 years you've matured enough and proven yourself to be trustworthy.  Four years when you're in your 40s isn't all that much, but when you compare an 16-year-old to when they were just 12, then that's huge!

If you have a genuine interest in photography, and you're taking this hobby seriously by working on your computer editing images, printing out your photos and framing them, etc..., I think your dad will probably notice and may offer you the camera without you even having to ask.

What are you doing for your photos now? I assume you're using your phone which is quite limiting compared to the D90.  In fact, there are many types of photos that the D90 can do far better than your phone and many your phone simply cannot do at all.  If the phone, or whatever camera you're currently using, is holding you back (e.g. no manual settings, no interchangeable lens, no continuous focusing, etc...) explain that to your father.

Of course, if all you want to do is try the camera out because it looks cool and stuff, then just ask.  If he says no, then live with it.  You'll eventually be old enough to afford a great camera in the very near future anyway.

The D90 is not obsolete by any stretch of the imagination.  If you think it takes worse photos than a modern phone, I would encourage you to check out these photos on 500px.com that were taken with a D90: https://500px.com/search?q=D90&type=photos  There's not a phone on the planet that can compare to the capabilities and image quality of even an obsolete camera like the D90.

I think your odds of success depend partly on your current age.  If you were 12 when you asked the first time, the only answer you were gonna get was gonna be "no."  So now you're 17 and adult and MAYBE if he thinks you'll use it and not beat it up.  If you were 22 then, it's gonna be the same questions and whether it'll just get stored in YOUR closet.   Ask him if you could borrow it for a couple of weeks to see if you like camara photography.  IF you like it, offer to buy it or trade something for it.  Or offer to buy him something to trade.  

You need to tell him you want to get into photography as a serious hobby, and if he could help you learn by lending you his D90. Even 6mp DSLRs could produce pictures WAY better than any tiny sensor smartphone today. Even the little s400 Canon powershot compact could.

Its a 13 year old camera. If the shutter count is low he will keep saying no. 

My 4 or 5 year old 16 MP Fuji compact camera is still fun and takes great photos.

The MSRP on that rig was $899 without the lens, so it would have been a considerable investment at the time, and probably his pride and joy (not counting you, of course). He'd have to feel done with it and that it would be taken care of, so I'm not surprised by a hard no back then. Still not super likely.

Idea: find a good "pro camera" app for your smartphone and learn the basics (composition, exposure, depth of field, etc.) that way. Some amazing work can be had just out of a phone. 

Ask him to teach you how to use it.

If he sees that you’re taking notice and learning he might give it to you or lend it to you. 

Once you start to learn, you’ll quickly find out why even a much older D70 can give images of FAR better quality than even the best current phone camera. A D90 is even better. 

Tiny image sensors are always beaten by large image sensors. High megapixel counts are NOT a measure of IQ (Image Quality).

You don't start out by asking him to give it to you.

You ease in by asking him to teach you to use it. You pay for film and developing (or do your own developing if you have access to a darkroom and some instruction as needed). Your focus needs to be learning how to use the camera, focus, lighting, composition and framing, like that.

Then if you're careful with your dad's camera and he's taught you enough, ask if you can borrow it. Or even rent it. Return it in perfect condition every time.

Eventually, when he sees you taking proper care and using it well, he *may* give it to you.

It would be a great camera to use. You could ask your dad to teach you how to use his camera or ask him if you can borrow it and try it out. If he says, no, he's just being selfish.