So, I read somewhere that when posting non-created generator pets (for a contest or whatnot) you are supposed to use a link that looks like "pet_builder" something something something, but if I try to get the image url from it I just get a string of letters and numbers and the image is very small. How do I get that other url?

If I'm understanding correctly, you want the image to be bigger?
If so, adding &size=250 (or any number between 50-500) to the url, changes the size of the image.
Ex:
https://www.sylestia.com/image_builder_pet.php?species=aeridini&gender=male&E=0&gd1=00&gd2=00&gd3=00&md1=00&md2=00&md3=00&ec=FFFFFF&c1=FFFFFF&c2=FFFFFF&c3=FFFFFF&c4=FFFFFF&c5=FFFFFF&gd1c=FFFFFF&gd2c=FFFFFF&gd3c=FFFFFF&md1c=FFFFFF&md2c=FFFFFF&md3c=FFFFFF&ac=FFFFFF&ac2=FFFFFF&maturity=216&size=250

Not exactly what I meant but that actually helped me figure it out, thanks! :]

the string of letters and numbers in a pet image url (or avatar image url) is the codes for all the colour slot hex codes and the trait letters, if you were wondering!

the &size= modifies the number of pixels in width of the image is, and is present on the small previews in the generator (set at 150). The large preview when you click on one of the small ones lacks the size modifier and thus shows at full size (500; the size modifier can't make the images lager than the full size so if you put any number larger than that it'll default back to 500/max).
For official pet/theme design threads, 250 is the typical preferred size (since it's a decent size for seeing details and fits two pet images next to eachother on one line in a post)