Drainage question

My house has a drainage ditch in the back yard that runs from east to west. Basically the drainage from a handful of houses to the east runs through my backyard to a 6in drain. My neighbor said he was around when they installed it and apparently it's a 6in drain that has a 90* coupler that attaches to a 12" pipe that runs under the rest of my property, under his property, under the street and drains into a field.

I've had flooding issues 2 out of the 3 years I've lived at this house because the 6" drain inlet doesn't drain the water fast enough to keep up. If it was a 12" inlet, I'm sure I'd have no problems.

I'm trying to figure out how I could somewhat cheaply get my drainage working better. Is there a way to dig down a few feet past my drain inlet and drill a hole in the underground 12" drain pipe and make a second inlet connection without using a tee coupler? That way I'd have two drains going into the same pipe? I don't think the couplers that big box stores would work on this drain pipe and if they did, they cost an insane $550 for a 12" tee coupler.

I'm no plumbing expert so I was wondering if you guys had any ideas?