Is it possible to repack an .apk to install it in an incompatible Android version? ('there is a problem parsing the package'/have 4.0.4-requires 4.1) It's about an .apk (Inbox by Gmail) that requires Android 4.1 and up.
Trying to install it in my phone which has Android 4.0.4
it shows that "there is a problem parsing the package".
I have enabled installing .apk from sources that aren't from the Play Store on my phone.

As I've read this is caused because the .apk requires a different device API level than that of the phone.

Is there a way to convert/repack the .apk so that it can be installed to my phone?
Short answer: No.

Simple repacking cannot solve the compatibility problem. If an app requires "at least Android XX", that usually means it needs some conditions not met before it. There are new OS features introduced with each Android version, libraries get additional APIs/features, etc. – nothing of that can be solved by "repacking" the installation package. If the (required) functionality cannot be provided by the Android version your device is running, the app cannot work (correctly) – even if you were able to install it.