So I think we are misunderstanding your issue. To clarify, you're trying to get your TIRES to lock up, this isn't an issue of the actual hand-brake handle staying in the up-position when the car is parked (which would be a pawl/detent issue within the lever mechanism.) Is that right?
I'd be worried about over-heating your rear brakes and long-term damage to your hubs from the pad constantly dragging like that, pumping heat into everything, not to mention premature brake wear.
I don't drift, I DO track and road-race my cars though, and my honest advice is to read up on what you're trying to do a little more. Drifting isn't going fast and just whipping the steering wheel and trying to get your tires to lock up. Suspension geometry plays a big part, your engine power, and how you balance that traction in the rear of the car. If you're trying to do this in a stock S13 with decent tire, fairly neutral camber, and an open diff, blown struts/shot bushings etc..., on a paved road you're probably not going to be getting the result you're looking for. I don't know what environment you're trying this in either but it's also entirely possible you're not carrying enough speed into the turn before you initiate. My S14 behaves completely differently in a corner than my 5th gen camaro does.
Short answer; there may be nothing wrong with your ebrake at all. It isn't a drift lever.
The stock e-brake is just that; an emergency brake to keep the car in place while parked. Some people use it to initiate a drift, sure, but it's only a small part of a much bigger equation in balancing rear traction for what you're trying to do here.
Not advocating for doing anything dangerous on public roads /disclaimer
You can use a combination of feint/clutch kick to initiate as well but this all hinges on balance, carrying enough momentum into the slide, and doing it in a safe place where you can learn and practice. I'm largely ignorant of the drift community but in NASA/SCCA stuff there are local events where you can come to participate. They'll pair beginners with a coach to both learn the track itself and proper safe technique as a jumping off point. It's a great community and if you can find something like that in your area I'd 100% recommend going that route.