I'm going to assume your house sets on somewhat of a hillside, and they cut a flat bench to build on.
That being said if that is the case, cutting that bench has gotten closer to the level the spring is perking out. Seeing what you have in mind to install, depending on how close to the surface the water is perking out may not solve the problem.
If you have a slope going uphill from the parking area, my thoughts would be to put in a horseshoe type french drain starting far left near the lateral you have, stepping into the slope maybe 18"-24", digging down to find the source of water above the parking area catching the water before it gets to the parking area. It'd be super helpful to dig down before and find the level of where water is at/perking off a hard surface.
I'm also going to assume there may be a layer of rock, or depending on your location a layer of fire clay, but that is usually only found in regions with coal. Water will soak down through dirt and coal until it hits the fire clay which is pretty much water proof, and perk out at that level. If no coal in your region, I'm guessing you have a layer of rock it is following out.
Catching the water before getting to the parking area, and long before the house will do what you want to do and dry things up. A 8" to 12" trench as deep as it requires with a fabric covered 4" plastic slotted pipe will carry away a lot of water. With seeing the trees alongside the drive and what may be out of the picture, just using gravel tree roots will go for the source of water and in 5 years or so may restrict/clog your drain.
Going with a trench 18"-24" deep will allow you to cover the pipe with around 12" of #8 gravel/pea gravel and catch any extra water that may perk out during the wet season. And you'll want to cover the gravel with some sort of fabric to keep dirt from sifting down through. On the job where we've installed drains on road slips we used surplus construction fabric contractors had left over, and gave back to us. Here at home, I've used the "better" 5 year landscape fabric used around tree's etc., folded over 2X. I dug into a drain I installed 15 years ago to connect more drainage, and the fabric I put in looked like new. Keeping the elements and air away from it will make it last a long time. Heck, years ago before fabric they used a layer of straw on leach beds to do the same thing. We did the same before using fabric. Some friends had an excavating business that installed septic systems. They would replace systems that people had planted trees and had clogged the laterals, and still found straw although somewhat deteriorated, still doing it's job.
You don't need a whole lot of fall 2% is a good amount of drop. That will flow water plus carry any sediment out that may get in the pipe. A lot of drop will let the water run fast and drop the sediment.
Once you get beyond the point of where you're picking up water, you'll want to go into solid pipe with no gravel only dirt. Do this right at the end of the perforated pipe and it will force water into the perforated pipe. Otherwise, you'll have water constantly seeping into the surrounding bank and possibly cover the outlet.
I'd definitely make some sort of splash pad or put larger stone in a trench at the outlet. You'll want that trench back a foot or so before the end of the pipe. Eventually, water will undercut the pipe when down to a trickle. Eventually you'll still have some erosion beyond where the splash pad/stone ends.
Whether you do it yourself, or contract it, having a good grade is most important. Having a belly in it somewhere along the line, water will lay there, and still perk out towards where you want to keep it out. And you'll want to check your outlet several times a year making sure you still have a flow, and check for erosion beyond the pipe outlet.
Another thought came to mind... If your County has a Soil Water Conservation office, contact them and have them have a look and maybe come up with a plan. The service should be at no cost. Your tax dollars pay for it, might as well get some benefit out of it. They will come out, assess the situation, shoot grade, etc. and come up with a plan, in blueprint form. Look it over then decide if you want to tackle it or let a contractor do it. If you do it yourself, a transit level and grade rod would be a must to keep proper grade. Just keep in mind a grade rod is in tenths of a foot, not 1/8" increments.
