Is following sentence grammatically correct?
The world is getting smaller and international connections tighter.
What kind of construct is this? Parallel structure?
In a sentence like
The world is getting smaller and the global economy more tightly integrated
the omission of ‘is getting’ in the second clause is fine, because the subject in each case is singular. Where one subject is singular and the other plural in a sentence such as the one the OP gives as an example, many readers might feel there’s something wrong. So let’s be pragmatic. Whatever the grammatical rights and wrongs, it takes only a little extra effort to write instead ‘The world is getting smaller and international connections are getting tighter’. That way you can be sure of getting your meaning across to the maximum number of readers without any of them being distracted by what may, admittedly, be a trivial grammatical point.
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".