Short answer. From notice of the denial of your motion. Where a motion for reconsideration or new trial is timely filed, the rule counts the sixty days from notice of the denial, and it says so whether or not the motion was required.
What the law says
In case a motion for reconsideration or new trial is timely filed, whether such motion is required or not, the petition shall be filed not later than sixty days counted from the notice of the denial of the motion.
Rule 65, Section 4 — When and where to file the petition. Read the full provision →
A fresh sixty days, not the remainder of the first
The rule does not suspend the original period and give you back what was left of it. It fixes a new reckoning point: sixty days counted from the notice of the denial of the motion. So a litigant who used most of the first period before moving for reconsideration is not left with a few days on the denial. The general rule in the same provision — sixty days from notice of the judgment, order or resolution — is what applies where no such motion was filed.
The condition is that the motion was timely filed
This is where the benefit is won or lost. The re-reckoning applies in case a motion for reconsideration or new trial is timely filed. A motion filed out of time does not answer that description, and a litigant who files late may find the period still running from notice of the original order while believing it restarted. Whether your motion was timely is therefore not a side issue — it determines which date your deadline is measured from.
It does not matter that the motion was optional
The rule adds four words that resolve a common worry: whether such motion is required or not. A litigant who moved for reconsideration where it was not strictly necessary does not lose the benefit of the later reckoning for that reason. The provision treats the timely filing of the motion as the trigger, without asking whether the mover was obliged to file it.
Count from notice, and note the source of the rule
Everything here runs from notice of the denial, not from the date the resolution was signed or the date you heard about it informally, so the record of when notice was received is the document that governs. Note also that this section carries an amendment notation in its own text — (As amended by A.M. No. 07-7-12-SC, December 12, 2007.) Because the consequence of miscounting is that the petition is out of time, confirm the operative dates with a lawyer rather than working them out from memory of when things happened.