I know this is a tough one, since there are so many family expectations of who should be invited, and since people take a wedding invitation very seriously as a sign of being loved, belonging, etc.
Since you are planning and paying for your wedding, and you want to keep it small, you have every right to choose who will be invited to the wedding. Parents should be told that while you understand their feelings and wishes, and you certainly don't want them to take any heat from extended family about who's in and who's out, it's important to you to plan your wedding according to your wishes. Given the size of your budget, and your wish to be surrounded only with the people who are closest to you, this is what you've decided to do.
Since you can't just state this and walk out of the room (not a good diplomacy move), why not encourage your parents to renew their wedding vows, and thus invite all of the people they want on THEIR list? Sometimes parents want to celebrate with extended family and their friends, so this happy suggestion does often solve this problem. And it gives them something to look forward to.
Feelings may be hurt, but in this age of smaller wedding guest lists, hugely expensive weddings, and bigtime stress in the planning of a wedding, you may find that the majority of the 'left out' guests will understand, as they have likely found themselves in this position with family and friend weddings of their own, and many express that they were in this same position when they were planning their own weddings. So while some people are going to grumble, others will be fine with it.
As for those with hurt feelings, they'll have to get over it, since no bride and groom should be forced to expand their wedding guest list over have-to's and parental pressure. Many guests accept the facts and will send you a congratulations card, and some will send a gift. Your thank you note for either is where you thank them for their good wishes and let them know you'll look forward to seeing them at (future family function.) If you stay confident in your decision, it will all go much more smoothly. Again, this is a tough situation, and you can't please all the people all the time. You must, though, stick with what's right for you. You'll be proud of yourself in the moment, later on, and you may start a welcome precedent with other brides and grooms in your circle of family and friends. I come from a big family. My grandmother was one of eleven siblings, so I have hundreds of cousins and great-cousins, etc. One brave bride years ago decided to just invite the relatives with whom she was close, and a small wave of grumbles turned into an acceptance: "Yes, this is a better deal." We've all planned our weddings like this since then, and everyone understands.
And for those who want to create a big drama....it may flare up, but they'll move on to creating a drama over something else in just a short while.
I must add this caveat...keep family diplomacy in mind, and make sure you're not leaving out people who really should be there. Some family ties are golden, even if you haven't had much contact with these relatives, so use your best judgment, and know that compromising with a few additions could smooth the road with your parents.
I'm sure the rest of our Pashers will add their insights...
All the best,
Sharon