What are the parts used for the [Hirolabo delta robot's] arms and circular base (labelled as parallel links and universal joints) ?
Let's see another still from that video:
There I see:
I'd like to use Mindstorms 51515 [...]
Well, I'm looking at the parts list for 51515 and am seeing no universal joints and only four towball axles. Seems like you'll need to replace each universal joint with two perpendicular joints.
Working with a limited selection of parts is always challenging, but I think 51515 has everything you need. Try something like this for the actuator and the base of the parallelogram:
The key is using 6536 Axle and Pin Connector Perpendicular to provide the extra degree of freedom. It's gonna be worse than a universal joint or a ball-and-socket joint, but it's what you've got. Remember to use the low-friction gray pins for the joints (instead of the high-friction black ones).
I've drawn only one arm, but the other one is symmetrical; and I haven't drawn the pins - add them as needed.
The top part would also use two connector blocks, and perpendicular-axle-pin-connectors for the arms:
Since 51515 doesn't have any parts with an hexagonal pattern, or any other easy way of achieving a 120° angle, you'll have to rely on interconnecting the upper assemblies (with 3L pins):
That still looks flimsy, so consider reinforcing the assembly with more interconnections.
You'll need to fiddle and test, but I think this design should provide you with a solid base to start tinkering with.
Also note that the joints in this design are lacking a degree of freedom, and that this might apply excessive torsion to the beams used in the six arms. Experiment with adding an additional 6536 perpendicular-axle-pin-connector to each joint to provide that extra degree of freedom.
Of course, you can always go shopping for more parts. In your case, you would only need twelve of 32474 ball joint and 93571 ball joint socket. You should be able to acquire those from the usual marketplaces (pick-a-brick, bricklink) for about 0.5$/0.5€ each pair, or about 6€/6$ the full set.
Welcome to the rabbit hole of "I just need to buy one more of that one piece".