What is a word that you would call someone if they only believe the parts of the truth that they want to believe?
It sounds like you might be talking about confirmation bias:
Confirmation bias [...] is the tendency to search for, interpret, favor, and recall information in a way that confirms one's preexisting beliefs or hypotheses. [...] People display this bias when they gather or remember information selectively, or when they interpret it in a biased way. [...] People also tend to interpret ambiguous evidence as supporting their existing position.
Wikipedia, excerpts removed for brevity
Confirmation bias is something that happens subconsciously. As per the same Wikipedia article:
Confirmation bias is often described as a result of automatic, unintentional strategies rather than deliberate deception.
When people only hear what they want to hear, we call them 'selectively deaf', i.e. they choose what they hear and what they don't.
Please note that, compared to the other answer about confirmation bias, that being called 'selectively deaf' implies that this person is doing this consciously, it is their intention to do so. Confirmation bias is subconscious.
I'm not sure if you are talking about someone being intentionally selective; my answer assumes that you are.
I can't find a definition for this, but googling "selective hearing" yields many results; this term is actively in use, though seemingly not strictly defined by a respected source.
For your case, it's the acceptance of reality (objective truth and reality are functionally equivalent, seeing something as true means accepting that it is real).
The attitude or practice of accepting a situation as it is and being prepared to deal with it accordingly.
A person who accepts a situation as it is and is prepared to deal with it accordingly.
So it's appropriate to call them a 'selective realist'.
I also thought of NVZ's suggestion: cherry-picker.
The OED reference (sign-in needed) copied below includes some interesting background on the origin of the term. It comes as slang from the U.S. Railroads back in the early 20th century.
cherry-picker 3. colloq. (orig. U.S. Railways slang). A person who selects only the best or the most profitable items, opportunities, etc. Cf. cherry-pick v.
(Before accusing me of cherry-picking the best definition, I feel in this case it is justified.)
People sometimes cherry-pick parts of the Bible that they want to believe.
Some examples of usage with dates:
1940 Railroad Mag. Apr. 40/2 Cherry picker, switchman, so called because of red lights on switch stands. Also any railroad man who is always figuring on the best jobs and sidestepping undesirable ones (based on the old allusion, ‘life is a bowl of cherries’).
1984 Supermarket News (Nexis) 2 July 16 He urged produce managers..to make shoppers of cherry pickers who, he said, come in mostly for the store's hot weekly specials.
1993 U.S. News & World Rep. 8 Feb. 62/1 The best of cherry pickers make decisions based on individual companies' attributes rather than broad market trends and place big bets on relatively few issues.
1999 Jrnl. Business 72 237 Further, cherry pickers (those who visit a store only to buy items on promotion) are less store-loyal.