I work in Staines and there's a fairly long stretch of Wraysbury Road here https://goo.gl/maps/iMQTWKTwdszfaWqi6 where there are often spaces, though it can be a bit tight in the mornings. No parking restrictions at all. Then it's £15 in an Uber to the airport, or get the number 8 bus from the town bus station for £2 (!!)

I've also parked in the past for a day-trip in one of the residential roads near Hounslow West tube station. Dene Avenue and Bulstrode Avenue have a lot of dropped kerbs but there are often spaces. Whether leaving a car there for such a long period is a good idea is another matter - you may understandably attract the ire of the residents if you leave a car there for weeks.

You can also park for up to 30 days at Hounslow West tube station car park. This is now our car park of choice - we're flying off today for 10 days and it's cost less than £60. Three stops on the tube and you're there. You need to read the signs and be careful to park in the right-hand side of the car park (as you see it from the main road), the left hand side hosts a car boot sale at weekends.