Unfortunately, Amazon does not allow tracking its deliveries with regular tracking numbers. That's why you need to use the original tracking link you received from Amazon.
Moreover, you won't be able to see delivery information straight away after adding an Amazon delivery from Web Access, you will still need the iOS or the Mac app (as described in step #7) to sign in to your account and download data for you.
Here is how to track a delivery with Amazon Logistics in Parcel Web Access.
You can get all of your Amazon deliveries automatically added to your Parcel app with Parcel Email service. Read more.
Alternatives If you are unable to add a delivery from Amazon this way, please try adding it using the iOS version of the app. Here is how to do that.
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