In general you should expect to see the following clearances on the parents of any puppy you consider. OFA (http://www.offa.org) clearances on hips and hearts. Hip clearances come in Excellent, Good and Fair. All three are acceptable for breeding. Hearts are clear of murmurs and SAS (Sub-aortic stenosis) or they are not. Your preference would be that the heart clearance is from a Board Certified Cardiologist rather than a Specialist or General Practitioner. An eye clearance must be from a Board Certified opthalmologist. Most breeders submit these to CERF (http://www.vmdb.org/cerf.htm) so that you can check their database for clearance information.
The Golden Retriever Club of America has recently recommended Elbow Clearances as well. Most breeders are doing this with current dogs; some are not. My impression is that for a few breeders this is a gray area now, as is the issue of whether a dog having Grade 1 elbow dysplasia should be bred (as it would be in all other countries). You should ask about elbows.
Not only do you want to know about parents, but about the siblings of parents and grandparents. The only dog in a litter that did not have hip dysplasia is probably not a good bet as the mother or father of your puppy.
A note on checking the above web sites for clearances. Always use the dog's registration number rather than the name as these databases are sometimes not able to pick up dogs by name.
There is a great deal of information available to suggest that overzealous owners and vets have been their own worst enemies with regard to their dogs' health. Current data indicates that minimum vaccinations are far better for our dogs. Many people have to do rabies due to governmental constraints, but I urge you to read the document below for the latest information. You will be given a copy of this for your veterinarian if you have a Sunbeam puppy.
