Do Jazz musicians make any effort to avoid parallel 5ths and octaves?
Short answer: no, no effort at all.
Avoiding these are rules of classical music and counterpoint, and they do not apply in Jazz. Quite the contrary in fact : it is very common to harmonize for a 5 sax section with the melody doubled 1 octave below the lead voice, and using e.g. a drop 2 or a drop 2 4 voicing. Same thing with trumpet/trombone sections in big band.
Check for instance http://www.timusic.net/debreved/jazz-melody-and-voicing-part-2/ (section "5 Parts")
In solo jazz piano, these are definitely not avoided altogether. It's not uncommon to see chord voicings that "double" a single note (e.g., octaves), although too much doubling can take away the texture of the other notes. Parallel fifths are also not avoided, but you're probably not going to see a ton of parallel fifth movement unless the chords are moving up/down the scale one scale tone at a time (eg, | C maj | D min | E7 | F maj |). Monk used 1-5 voicings in the left hand frequently, which could in some cases spell out parallel fifths if you were to analyze just his left hand in the lower register.
If you consider modern jazz piano in a group setting (with bass, drums, and a horn), you'll see basically the same thing on parallel fifths but much less doubling of notes. So octaves are much more likely to be avoided in that context when comping behind a band. There are exceptions, but this is a general rule of thumb that some jazz piano teachers state explicitly to new students.