What was the temperature in the shade of the Parker solar probe? Do to the proximity at the closest point the probe got to the Sun how much shade can be produced at that range?
At Parker Solar Probe’s closest approach to the Sun, temperatures on the heat shield will reach nearly 2,500 degrees Fahrenheit, but the spacecraft and its instruments will be kept at a relatively comfortable temperature of about 85 degrees Fahrenheit.
Source is here, admittedly it's far shorter than I had hoped.
In addition to the quote stating the approximate temperatures, to respond to how big the umbra of the shadow is... the heat shield is 8' in diameter. Basically, this heat shield has unimaginably low absorption rates, so while it does absorb some of the sun's rays, a large amount (99%+ I don't know the exact nubmers) are reflected away from the components.
It's not about how much shade can be produced, the ray scattering of the sun would eventually close the "shade" back up behind the probe. This shade could apply to the probe, and anything in a conic area directly behind the probe. The only exposed parts are the solar arrays, which are only partially exposed and some scientific instruments, which are required to be exposed for measurements. These are also made of a non-average materials to withstand much of the direct exposure to the sun.
Another good source of information can be found here. It explains more in detail using a video how the Parker Solar Probe will mitigate much of the heat.
A few quotes from that page pertaining to the question:
Of course, thousands of degrees Fahrenheit is still fantastically hot. (For comparison, lava from volcano eruptions can be anywhere between 1,300 and 2,200 F (700 and 1,200 C) And to withstand that heat, Parker Solar Probe makes use of a heat shield known as the Thermal Protection System, or TPS, which is 8 feet (2.4 meters) in diameter and 4.5 inches (about 115 mm) thick. Those few inches of protection mean that just on the other side of the shield, the spacecraft body will sit at a comfortable 85 F (30 C).
As for some of the instruments that will not be protected by the shield fully:
Poking out over the heat shield, the Solar Probe Cup is one of two instruments on Parker Solar Probe that will not be protected by the heat shield. This instrument is what’s known as a Faraday cup, a sensor designed to measure the ion and electron fluxes and flow angles from the solar wind.
The cup itself is made from sheets of Titanium-Zirconium-Molybdenum, an alloy of molybdenum, with a melting point of about 4,260 F (2,349 C). The grids that produce an electric field for the Solar Probe Cup are made from tungsten, a metal with the highest known melting point of 6,192 F (3,422 C). Normally lasers are used to etch the gridlines in these grids — however due to the high melting point acid had to be used instead.
Please note, I am by no means an expert and am paraphrasing things I've read in the non-quoted sections. Let me know if the information I've provided is inaccurate. However, the block-quotes are direct excerpts from NASA, and sources are provided.