I've seen many cages, particularly the small cages, have guillotine doors. Guillotine doors are the kind that slide up and down, and they are gravity fed. Basically, unless you put locks on them, there is probably nothing to stop a sneaky or curious little bird from pushing up the doors and getting out. The problem is that many small bird cages are poorly made and cheaply made and it's difficult to find a small cage that has suitable bar spacing for a small bird without these guillotine doors.
It's difficult to explain but this is what I mean by guillotine doors. The half circles on the front of the cage are where the doors are. And these doors slide up where a food bowl would be inserted.
This is what could happen if your little parrot figures it out when you're not home (see 1:18):
Are there any small cages for small parrots that don't have guillotine doors offered on the internet? I've been looking and I haven't found them. There doesn't seem to be much of an alternative if the bar spacing is too large (particularly for a budgie). So, beware of guillotine doors, and put metal locks on them.