When my Mac is rebooted, it arrives at the screen with my UserID. When it is like that, I cannot access it remotely-- the remote app says the Mac is offline. Once I log in at the Mac itself, then the machine shows online and I can access it remotely.
WiFi connection. This is normal behavior.
Because your Mac is not connected to Ethernet cable, so when Mac reboots no network connection is available until you login and get WiFi connected.
This is principle how Parallels Access works.
WiFi is connected using certain WiFi network and certain credentials (user/password). Each host user on Mac can have different WiFi network and credentials. When Mac is rebooted - there is no current logged in user - so no WiFi network and credentials are available. This behavior is by design.
NOTE: Also, FileVault enabled in System Preferences is blocking WiFi connection. You may disable it and check how Parallels Access will work.