Aaron’s grandson Pinchas is rewarded for his act of zealotry in killing the Simeonite prince Zimri and the Midianite princess who was his paramour: G‑d grants him a covenant of peace and the priesthood.
A census of the people counts 601,730 men between the ages of twenty and sixty. Moses is instructed on how the Land is to be divided by lottery among the tribes and families of Israel. The five daughters of Tzelafchad petition Moses that they be granted the portion of the land belonging to their father, who died without sons; G‑d accepts their claim and incorporates it into the Torah’s laws of inheritance.
Moses empowers Joshua to succeed him and lead the people into the Land of Israel.
The Parshah concludes with a detailed list of the daily offerings, and the additional offerings brought on Shabbat, Rosh Chodesh (first of the month), and the festivals of Passover, Shavuot, Rosh Hashanah, Yom Kippur, Sukkot and Shemini Atzeret.
I am looking for a source for the upper limit of the counted for the census. The verses clearly say from age twenty and above, but I cannot find the upper limit of sixty that you mention here.
There is actually a difference of opinion in the commentaries on whether there was an upper limit of sixty.
This short synopsis seems to follow the commentary of Rabbi Moshe ben Nachman (Nachmonidies) on Numbers 28:64 that above sixty wasn't counted. This would partially be based on the verse (ibid 26:2) " Take a census of the whole Israelite community from the age of twenty years up, by their ancestral houses, all Israelites able to bear arms." i.e. they only counted those who would bear arms which is between the ages of 20-60.
The Ramban on 26:64 notes that the count was only until Sixty. There he also cites the Talmud Bava Batra 121b where there is an exegesis from Vayikrah 27:7 that clarifies this point.