All of the -stan countries were under the Achaemenid empire (known as the first Persian Empire, covering Afghanistan, Pakistan, Tajikistan, Kazakhstan, Uzbekistan, Turkmenistan, Azerbaijan and Kyrgzstan. This was in 550-330 BC,so the Persian language was being spread across Central Asia.

The Samanid Empire, which was Persian, covered Kazakhstan, Pakistan, Afghanistan, Tajikistan, Kyrgzstan, Uzbekistan and Turkmenistan. They ruled for approximately 180 years, from 819-999. It was after this that Central Asia and the -stan countries of Central Asia began to have more Turkic influence. The shift to Turkic cuiture was with the Kara-Khanid Khanate, ruling in Central Asia from 999-1211, but they gradually assimilated to the more Persian influenced countries.

Aside from empire, there was also the Silk Road. Many Iranians (not necessarily Persian speaking) travelled and traded along the Silk Road, for example, the Sogdians, who spoke Sogdian, an Eastern Iranian language derived from Indo-Iranian. Sogdian influence was heavy along the Silk Road, from China all across the Silk Road,continuing up until the 10th century. They were the main traders for the 3rd-5th especially, so that would mean a lot of Iranian influence.

Overall, all the -Stan countries had a huge amount of Persian and non-Persian Iranian influence. To describe them as not being Persian countries now is correct, but historically they were being passed from one Persian empire to another.

Keep in mind that every country or region ending in "Stan" has or had a strong Persian influence. It's similar to how many former British colonies or regions end in "land".

Note that the name of the Indian state of Rajasthan contains the Indo-Aryan cognate of the Iranian -stan.