Congress mandated that NASA construct a surveillance capability for relatively-large NEOs (larger than the ones in OP), so the detection problem has been well-studied. A little googling will turn them up, but here's what I remember:
- Detection in optical by ground-based telescopes is a good technique. You need several exposures and good spot-tracking. Because of the relatively short time-base, you also need good follow-up or else the NEO diverges from your estimated orbit and is lost again!
- The parallax issue you mention comes up more regarding orbit determination than detection per se. (I.e., you need more than just detection to do anything useful.) One link going deep on the relationship of parallax to good orbits: [1]
- There are a bunch of such optical surveys, e.g. Pan-STARRs [3].
- But ground-based telescopes rely on reflected light, and so NEOs coming from certain regions (e.g., within Earth's orbit, but it's worse than that) don't have much reflected sunlight so are hard to detect optically.
- So, IR telescopes can be used, but you have to go above the atmosphere. See, NEOSurveyor [2]. IR is based on the infrared "glow" of a NEO against the background, and so it does not rely on reflected light. So IR telescopes get around many of the geometric limitations of optical.
- What about radar? It's very useful for follow-up orbit determination, but not so good for surveys, because the radar energy diffuses so much. Crudely, if the NEO is R away, you get a 1/R^2 on the way out, and a 1/R^2 on the way back. It's more complicated than that, but this is the reason that radar has limitations for this problem.
[1] https://iopscience.iop.org/article/10.1088/1538-3873/ac43ca/...
[2] https://www.jpl.nasa.gov/missions/near-earth-object-surveyor
[3] http://legacy.ifa.hawaii.edu/research/Pan-STARRS.shtml