HP does not provide official support for LaserJet 1012 on Windows 7. There is a workaround and it does work fine for me.

The solution is original from this post (http://forums13.itrc.hp.com/service/forums/bizsupport/questionanswer.do?admit=109447627+1270533366367+28353475&threadId=1381135).

1. First off, go download HP LaserJet 1012 Host Based drivers.
64-bit: http://h20000.www2.hp.com/bizsupport/TechSupport/SoftwareDescription.jsp?lang=en&cc=us&prodTypeId=18972&prodSeriesId=377934&prodNameId=306510&swEnvOID=2100&swLang=8&mode=2&taskId=135&swItem=lj-67501-1
32-bit:
http://h20000.www2.hp.com/bizsupport/TechSupport/SoftwareIndex.jsp?lang=en&cc=us&prodNameId=306510&prodTypeId=18972&prodSeriesId=377934&swLang=8&taskId=135&swEnvOID=2093
2. Extract the driver.
3. Plug in and turn on your printer if you haven’t already.
4. Go to Start>Devices and Printers, and choose to add a printer
5. Choose to add a local printer
6. Choose Dot-4-usb printing support
7. Click Have Disk on the next window
8. Navigate to where you extracted the drivers, and choose the inf file
9. Choose the HP 1012 HB Driver
10. Follow the prompt till it installs the printer or what not

Now you may notice that it prints slow as heck, well no worries! To fix that do this.
1. Go to Start, right click Computer and choose Manage
2. Go to Device Manager
3. Find 1284.4 dot4usb Printing Support
4. Right click it and choose Update Driver
5. Choose Browse my computer for driver software.
6. Choose USB Virtual Printing Support
7. Follow the prompt, exit out of Device Manager
8. Go to Start>Devices and Printers
9. Right click HP 1012 HB, choose Printer Properties
10. Go to the Port Tab, find Virtual USB Printing Support (Not dot4) and check that.
11. Click Ok.
Congratulations! Your printer isn’t an agonizing experience anymore!!
NOTE: The users in my environment reported that the printer after a restart showed up uninstalled but continued to print fine, so meh. Guess it cannot be perfect.