There is no need to log in to individual accounts, at least with more modern versions of OS X. You can set access privileges for managed accounts from within the Users & Groups pane of System Preferences without leaving your own account.
By selecting the Other tab in the Parental Controls dialog you can even enable Remotely managing parental controls.
Upon further clarification by Cra, it seems that Apple's file quarantine system is causing the difficulty. Starting with Mac OS X 10.5, files downloaded from the internet are quarantined and checked for safety.
Applications that are downloaded must be approved the first time that they are run, and only the owner of the account used to initially download the software can grant that approval.
If you have multiple user accounts on your Mac, the user account that downloaded the file is the only user account that can remove the quarantine attribute to the file. All other user accounts can open the quarantine file, but they will be presented with the quarantine dialog box asking "Are you sure you want to open it?" every time they open the file.
ᔥ Apple Knowledgebase:About file quarantine in Mac OS X v10.5 and v10.6
Make sure that you download software updates using your administrator account and that you open and approve the software from your account before making any adjustments (which may not be necessary at this point) to the Parental Control settings. It may be necessary to disable automatic updates in Firefox.
The quarantine system works by attaching an extended attribute to a downloaded file. If absolutely necessary, you can remove this attribute using the terminal
Enter the following command, replacing appropriately:
xattr -d com.apple.quarantine /applicationPath/<applicationName>.app
See also: Stack Overflow: How do I remove the “extended attributes” on a file in mac