Interfacing DS1302 - is the schematic I found correct?
Hobbyists don't always get the details correct, if it works they are happy. Follow the vendor recommendations.
The pull-ups are not only not needed, they can potentially cause harm since the DS1302 has internal pull-downs. I would guess that the author was accustomed to seeing pull-ups for I2C interfaces and thought that this was similar. The DS1302 doesn't use I2C or SPI, it is a custom interface.