I had this issue with my wedding, with parents insisting that work colleagues be invited, and also that far away relatives I had never met be invited (but don't worry, they won't actually come, it's just rude not to invite them) and also that certain people not be invited because they didn't like them. I saw my friends deal with it too, and it caused a lot of crying.
The problem with sending a "save the date" is that it's so early you don't yet have a venue and don't know your numbers. What I would suggest you say to your parents is this:
We are sending "save the date" to our core attendees. Some people can't make it to a wedding because they have conflicts. These cards are just going to the people we really must have there, our own close friends and close family. Once the venue is selected and we know the guest limit, we'll make a detailed list, and we'll send invitations to some people who didn't get save the date cards. We can talk then about whether there will be room for people we're less close to, like your friends or some far away family.
In some sense this is just kicking the argument down the road; you'll have to have it again later. But later, you will have more specific points to argue than "hurt feelings" and "make the effort". You can say look, the venue holds 120. We have 124 on this list. We have to drop 4 people. I think it should be [godparents] and [some other couple.] I know you would like to have them there but there just isn't room.
Alternatively, if your parents are paying, you can say:
If you want to include people we barely know, we will just not fit in that 120 person venue we were considering. Can you cover a 200 person venue? That will give lots of room to include everyone.
You may feel that this makes your wedding less intimate, more of a chance for parents to show off and less of a chance for you to celebrate with your friends and close family. If so, try:
If you want to include people we barely know, we will just not fit in that 120 person venue we were considering. But moving to another venue will change the "feel" of the wedding from the small intimate gathering we wanted. Can we keep it small and exclusive, and explain to [godparents] that our preferred venue just doesn't enable us to invite everyone?
Having now been part of a wedding in Covid times, that had about one-tenth the guest list that had originally been envisioned, I can assure you and your parents that you will be just as married no matter who is able to attend, and that there are other ways to celebrate and support a marriage than attending the actual ceremony.