The best way to prevent this is to figure out your budget, and how much you want to spend restoring your collector car. Next make a list of parts that will need to be replaced, and do some simple online price shopping. You will want to add in the labor costs of the work, if you will not be doing it yourself. Doing restoration work yourself will keep the restoration costs down considerably.
Used Classic Car Parts
Thanks to the internet, it is a lot easier to find classic car collectors who are selling used parts form their collector cars. They may have upgraded that part, are parting out an old junker, or are just plain strapped for cash and are parting with their collection. Before the internet, a restorer had to rely on calling junkyards or looking at trade magazines. Today, you can find many parts on ebay, craigslist, or other classic car classified ad sites. Another useful tool are forums revolving around your classic car. There will be many collectors in the forums that will be willing to help you locate those hard to find parts, or at least give you suggestions of where to look.
An advantage to looking for a used part is price. Generally a used part will be cheaper than a newly manufactured part. A disadvantage may be the wear and tear of the item, since it is used, and you may have to spend more time and money into restoring that part. You will need to do a cost/benefit analysis to see if this is worth it. If the part is not available newly manufactured, such as an old headlight cover or body part, then you probably don't have much of a choice. Another caution you should look out for is safety. Older classic cars do not have safety features like today's cars, and you may want to look at newer designed brakes, steering wheels/columns, etc. instead of used parts. Used parts may be great for body and interior parts, but you may want to think twice on safety and drive train parts.
Newly Manufactured Classic Car Parts
As stated earlier, a big advantage with newly manufactured parts is safety. Performance is another advantage, since a newly built transmission or engine parts may be built stronger than they were in the past. That of course is not always the case, but it may also be cheaper to repair or replace a transmission with a new one rather than rebuild a decades old transmission or other drive train part. Some classic car restorers pride themselves on restoring their car with original parts, so you will want to keep this into consideration. It may even elevate the value of the car a bit having original parts. The choice is yours.
Collector Car Clubs
Car clubs geared toward certain models are great ways to meet other people who may have worked on the same model, and may even have parts. If they don't have the parts, there's a chance they know the person who does. Car clubs are fun, and can be very important in large restoration projects. Maybe you won't find the part you were looking for, but you may find the knowledge of how to repair that part, or another way around your problem. More than likely, someone has run into your problems restoring the car.