LizG - I have read about your problem and the other posts with interest. Evidently whilst disabled car parking spaces are enshrined in law, Parent and Child ones are only "courtesy" ones which the supermarkets do to encourage mums to shop there. There are some wonderful Parent and Child threads on Mumsnet which make interesting reading.
I do have a blue badge - I applied for it a year after I had my left knee replaced. I do think that how easily the Councils let you have them is a bit of a postcode lottery, as I found it very easy. I had a word with my GP about my inability to walk and about the fact that I needed to be able to open the car door fully in order to get out, filled in the form and sent it off to the Council. The badge arrived shortly afterwards. I do not use it to park on the road where there are double yellow lines, just when I go to a supermarket, or similar. Like you I have also been stuck when I have used an ordinary space and then come back to find that the next car had parked so closely, I couldn't get into my car.
When I was on crutches after my operation (and before I had my badge), I just went into our local Sainsbury's and explained the situation to them - they were happy for me to use the disabled spaces. Good luck!