Which Yuga did the sanskrit grammarian Panini live in according to Hindu tradition?
Pāṇini lived in the Kaliyuga as per Hindu tradition. Chapter 179 of the Pratisarga Parva (second khaṇḍa) of the Bhaviṣya Purāṇa throws light in this regard:
आसीत्पुरा कलियुगे पितृशर्मा द्विजोत्तमः। वेदवेदांगतत्त्वज्ञो यमलोकभयान्वितः॥४॥
Earlier in the Kaliyuga lived the foremost of Brahmins named Pitṛśarmā. He had knowledge of the Vedas and Vedangas and was afraid of committing sins (of Yamaloka). (4)
चत्वारश्चात्मजाश्चासंश्चतुर्वेदैक्यधारिणः॥२७॥ ऋग्यजुश्च तथा साम तुर्यश्चासीदथर्वणः।
He had four sons who were knowers of the four Vedas and their names were Rik, Yajus, Saman and Atharvan. (27b and 28a)
पाणिनिः सामनस्यैव सुतोऽभूच्छब्दपालः॥२९॥
Saman’s son was Pāṇini who was a grammarian. (29b). Gitapress translation says: who excelled in the field of grammar.
Thus we can see that Pāṇini was the grandson of an eminent Brahmin of the Kaliyuga named Pitṛśarmā, and accordingly his life is to be placed in the Kaliyuga only.