I tried with my little digital voice recorder but it captured nothing.
What do you mean, “nothing”? I assume the noise is just not that loud? Did you try the Amplify effect in Audacity?
If you can hear it, you should be able to record it. But… Your ears & brain are able to “locate” the “live” sound coming from one direction while ignoring background noises from other directions. When it’s recorded and played-back through one or two speakers, the signal-to-noise ratio can seem worse and it’s harder to isolate the sound you want to hear over the background noise. (You’ve probably noticed that if you’ve done any audio/video recording with a video camera or smart phone… The background noise always seems when playback the recording, and you may have not even noticed it while you were recording.)
Do I need to rent a parabolic microphone?
No… That won’t be much different from getting the microphone close to the fan while recording.
If it’s low-frequency noise, you need a woofer/subwoofer/earphones capable of reproducing bass. It should be in the recording but you won’t hear bass frequencies on laptop speakers, or a small speaker that might be built-into your digital recorder.
If your voice-recorder is very small & light, you may be able to tape/tack it to a non-rotating part of the fan.
The vibrations will then be transmitted through the case of the recording device, (in-contact with the fan), to the microphone inside.
The deep sound on the recording should be visible on the spectrogram, but will only be reproducible on devices capable of bass, (so probably not audible on laptop speakers).