I have seen a board and I wonder to know what would be the advantage of having capacitor in parallel with fuse.
The capacitor is placed for protecting fuse in switch on/off system. Most loads are inductive such as motors and light bulbs, and in switching on or off, they will make a huge voltage from inductor equation, in switching on, this voltage will cause a huge current in a very little time, this can break your fuse without a SC, etc By using this capacitor (typically ceramic) you can filter this current and protect your fuse from damaging from this current.
Are you sure they are capacitors and not some polyfuse or thermistor?
This doesn't look like your standard manufacturer bodge, but more like an amateur fix which may explain why there are more questions than answers.
Soldering to a fuse is a tacky way to attach something across those two points. Why not solder to the holder or the pins on the reverse side?
Cheap small fuses are often secured to the element inside by nothing more than solder and the end caps by adhesive. In my experience, even briefly, soldering to the end cap causes the adhesive to fail and may reflow the internal soldered joint potentially upsetting the element causing the connection to break. Not all the time but enough that i consider this technique janky.
Also, is the user expected to solder a "capacitor" to the new fuse if the old one blows?