In the early days of the iPhone SDK push notifications were quite small. The entire payload was limited to just 256 bytes until iOS 8 was introduced in 2014.
Today push notification payloads can be much larger. How big depends on which version of the APNS API is being used to send the messsage
|APNS API||Payload Size Limit|
|HTTP 1.1 API||2KB|
|HTTP/2 API VoIP||5KB|
WWDC 2015: What's New in Notifications
Sending Notification Requests to APNS
Most, but not all 3rd party services that send push notifications support the newer HTTP/2 API that was introduced in late 2015. The newer API is more reliable and has more benefits for your apps than just the payload size. It's worthwhile to check wether you are using the new API or the older one.