Good old elbow grease to start with.
My computers often end up looking like this eventually without maintenance (this one was a rescue, so she didn't get the same sort of care my regular systems get)
, so I'm pretty familiar with cleaning out icky desktops.
Firstly, dismantle stuff as much as possible - passive components need more total cleaning than the active ones. You might want to take the opportunity to change out your thermal paste at the same time (since its a good excuse to!). I use qtips and toothbrushes to scrub out as much dust as possible from heatsinks and fans -e specially the edge of the fan blades and places where dust is built up.
Then use wipes for cleaning electronics for the rest. In theory, you could also wash heatinks in clean, sudsy water, then dry them out completely, if you remove the fans - I normally don't bother since it takes a while for systems to reach this state, and it takes a while to get that bad.
Once thats done, don't let the dust build up. Its easier to get rid of dust thats not crusted out - get a filter to keep the dust out in the first place, and do regular preventive maintenance every few months.
And of course, keep your computer switched off and covered while they are cooking (or better yet, as far away from cooking as possible), and convince your parents in investing in a cooker hood, so the grease is stuck there, not your PCs.