The present participle can be systematically deduced from a principle part, namely the infinitive. In pronunciation, you just append ing. In writing, it is a little bit more complicated due to the peculiarities of English orthography, but it’s still systematic. For example, you have be → being, go → going, make → making, say → saying, occur → occurring.
Therefore, once a student knows the infinitive, they can also form the present participle. There is no need for them to memorise it separately for each verb. This however is the defining nature of principal parts. Hence, the present participle is no principal part.
By contrast it is not possible to derive the simple past from the infinitive. There is no rule that captures take → took, go → went, make → made, and like → liked. Hence, students need to memorise the simple past for each verb (or at least the many verbs whose simple past is not simply formed by adding ed to the infinitive). Hence the simple past is a principal part.