Apple - How can I hide **some** birthdays in iOS Calendar?

You could edit out the birthdate information from those contacts you do not want to see in the Birthdays calendar.

Another way to accomplish this would be by manually adding the birthdays you do want to see to a new or existing calendar and then disable the Birthdays subscription.

As a compromise, you can disable the notification for that person's birthday. In calendar, double click on the birthday even and then change the alert to none. You will then be prompted if you want to change this event or all future events. Select all future events and you will no longer receive notifications for that birthday. It will however remain in your birthday calendar, but you will no longer be pestered with reminders (which is what I came her looking for before figuring out I could disable the reminder)