If a parent wanted to file for his unmarried child which he will be 21 years old next month in October can we still file now?

He can file at any time but as soon as the child turns 21, it will no longer be an immediate relative petition and the processing times will be much longer.

Yes, but it depends on what you are trying to accomplish. Is the parent a citizen or resident? Where is the child now? If the parent is a citizen, then the child would be an immediate relative until they turn 21. You can ask immigration to expedite it, but it is unlikely they can do it all by October. When the child turns 21 they will automatically change categories and have to wait for a priority date. However, it never hurts to go ahead and apply to establish your priority date.

Yes. But immigration must receive the I-130 petition before he turns 21 years of age in order to still be considered a child under the Child Status Protection Act.

It depends on the status of the parent. If the parent is a US citizen, as long as the child is under 21 at the time of filing, the child retains his classification as a child until the visa is granted to him. If the parent is a permanent resident, the child will roll over to the category of unmarried son or daughter of a permanent resident (F2B) once he turns 21.

If the parent is US citizen, file the petition before the child reaches 21 will preserve the child's status as a "child" so he does not have to wait for the quota.

You omitted some important information in your question. Is the parent a U.S. citizen or a permanent resident? If a citizen, when did he become a citizen - before or after the child turned 18? If before, the child might be a U.S. citizen by operation of law. In any other case, assuming that the child is in the U.S. legally, or can adjust status under Section 245(i), or is not in the U.S., the parent should file an immigrant petition as soon as possible, even though the petition will not be approved in a month (it takes from 5 months to 2 years). Petitions for children under 21 are processed faster. After the child "ages out", if the parent is a resident, the case will simply be transferred into a different priority category, from F2A to F2B (and after the parent becomes a citizen, he should notify the National Visa Center that the case should be switched to F1 category).