Depending on how long you choose to stick around in Bloodborne's Hypogean Gaol, you could encounter Darkbeast Paarl very early, midway through the game, or not at all. The first thing you'll think when you encounter this horrifying mass of bone and sparks is, "Nope."
This boss is a lot scarier than she (unconfirmed) looks. She's probably the most similar boss to other Souls bosses, being a big four-legged beast. Even if you find her close to the beginning of the game, the strategy outlined below should make short work of her.
Start Off By Staying Beneath Her
The best method for beating Paarl is to stay beneath her. She darts around the arena, spraying lightning at you like a lunatic, so under her is honestly the safest place to be. She only has one attack that can get you under there, an area of effect lightning ball that is telegraphed pretty clearly, so you'll have time to make a quick escape. When she starts to slowly move upwards, run.
For all this beast's might, her bones seem quite brittle. You'll be able to knock her down several times if you focus on hitting her legs. You should not lock on for this fight, as that will direct all your attacks at her center of mass. Hit her legs while you're underneath her and she'll come crashing down, allowing you to get several hits in while she tries to get back up. Like Cleric Beast, each of her limbs has some invisible health bar, so focus on one or use sweeping attacks that hit a few at once until she drops.
Pick Your Weapons
Heavy weapons work really well against Darkbeat Paarl as they can stagger her quite frequently. Lighter weapons will do too but you'll have to get more hits in with them. If you're using a heavy weapon like the Hunter Axe, Kirkhammer, or Ludwig's Holy Blade, charge their R2 attacks fully to stagger the beast quickly.
Sometimes, Paarl will disengage and run off to recharge her electricity. It'll be pretty obvious when she does. She'll rear up on her back legs and roar and spark. You can punish this if you're fast enough by charging after her to attack, interrupting her recharge.
Like all beasts, Paarl is weak against fire, so dousing her in oil and throwing molotovs at her or imbuing your weapon with flame bloodgems and fire paper is a great way to deal extra damage. She's also weak to arcane damage, making Ludwig's Holy Blade great for this fight.
Henryk's Hunter Set is good armour to wear when fighting this beast as it has high bolt resistance, meaning her lightning won't hit you nearly as hard. Your best bet is to try not to get hit at all though, and staying under her will really make that possible.