Its a 2 man job
Gentlemen, don't try to put the top up by yourself. It is best to do it with 2 people, one on either side of the car to prevent damage. Some suggestions....if the top refuses to open due to prolonged storage in a collapsed position, you may have to use a penetrating fluid on each of the pivot joints to loosen them up. Don't over do the lubrication, as you don't want to stain the top material. You may have to get a partner to help you open and close each pivot point several times together on opposite sides of the car at the same time. Go slowly and don't force the issue. It may well be that the top material has been stored for so long in damp conditions that it indeed may have shrunk and will not allow you to open it fully. If that is the case, cut the material off of the frame. Buy a top rebuild kit from one of the Thunderbird suppliers. It includes all of the pins, bolts, anti rattle washers and sundry parts needed to replace every joint. Then take pictures as you disassemble the top, one joint at a time and catalogue everything you remove. Sand, prime, and paint the entire frame assembly while apart. You may want to do the clean up and sanding yourself, and farm out the painting to a good bodyshop. Check before painting to ensure that the swing arm will be the correct beige colour, and everything else gloss black. Reassemble using all of the new hardware. You may want to replace all or some of the weather stripping. Some of it will rejuvenate if you massage liberal amounts of weatherstripping silicone into it with your fingers. It could save you lots of dollars. Then spring for the best top material kit, again from a Thunderbird supplier (Stayfast, Everfast or Colour fast, I can't recall the name). The best don't shrink when wet. Then get a shop who know little birds to install the new top. They can do it faster, and cheaper than some shop that hasn't ever done one before, and with consistently better results. They will need your car with the frame installed in order to do the job. This whole adventure is not cheap, but if properly cared for, the top will outlive your ownership of the car. Just remember to never store it wet or damp, and always have 2 people to put it up or down. This is particularly true of the early 55 top frames, which can pinch and tear the top material if stored just once improperly. I found out the hard way. Good luck!
