On the laptop of my parents OS Freya 0.3.1 is established. I live in other city, but I can be connected to them through AnyDesk therefore I have downloaded him Elementary OS 0.4.1. Tell please how to update OS Freya 0.3.1 to Elementary OS 0.4.1 without having physical access to the laptop but only through remote access of AnyDesk
You can't. Since you need to install the new OS from a bootlable usb.
I recommend you to wait a while until Juno is released and then install Juno instead of Loki.
The only thing I imagine you could try is to create the bootable usb from your parents laptop and then guide them trough the installation by the phone or videoconference.