grammaticality - Is the sentence "The world is getting smaller and international connections tighter." grammatically correct?

Yes, it's completely grammatical and is an example of ellipsis just as:

Daniel plays tennis while Jonathan and Andrew [play] rugby.

My parents went to Spain and so might my sister [go to Spain].

As exemplified here, ellipsis can occur even when the precise declined forms that the implied words would take if they were expressly included are not necessarily the same as the precise forms actually surfacing in the utterance. So for example, the form "went" is expressed, but the implied form of this verb is the infinitive "go".