There is no reason to inform your current work place. Providing that hours don't clash, and you have a minimum of 11 hour gap between both jobs you're fine.
I too have 2 jobs. Been doing it since 2006, I have not informed HR/managers etc at both jobs. Don't need to and won't ever give them that information.
I had 2 jobs for a while, but it was nothing like the situations being discussed here.
I was working for Royal Mail as a driver while doing The Knowledge to gain my London Black Cab driver’s licence.
When I ‘passed out’, I rented a Black Cab, and would drive to the Nine Elms, SW London depot where I worked.
The shifts consisted of 6hrs 36 mins, and starting times were weird, you could start at 05.19, and finish at 12.55, which included a total of one hour of breaks, or at any time, e.g. 11.17 and finish at 18.53.
As soon as I clocked out, I’d put the TAXI light on, and head over Vauxhall Bridge toward Victoria and earn a few quid in 4 or 5 hours, then go home.
The depot management knew what I was doing, but as long as I did my job there, it was okay, it was the guys I worked with who weren’t happy, it got up their noses that I was earning more than them, and when two one day strikes were called for some obscure reason, they wanted the Union of Communication Workers to sanction me for working the cab instead of picketing the depot gates.
After about 6 months, I came to realise what I could pull down with the cab, and that working for Royal Mail was impeding that, so I gave in my notice and quit.