Is there a word for someone who's skeptical only of statements that challenge their assumptions, and credulous of those that support them?
I think of a skeptic as one who thinks critically of whatever statement they encounter (looking for alternate explanations, questioning sources, etc.)
Is there a different word for unevenly applied skepticism, e.g. if I am skeptical of things which challenge my worldview, and credulous of things which support it?
A biased person has a lopsided view of facts, and interprets them according to their interest.
biased - "having a preference or an inclination, especially one that inhibits impartial judgment"; "a particular tendency, trend, inclination, feeling, or opinion, especially one that is preconceived or unreasoned"
lopsided - "leaning to one side"
This doesn't quite capture the whole sense of a "one-sided skeptic", but would cover many instances of a partial and preferential diluted sort of skepticism:
hypocrite
a person who acts in contradiction to his or her stated beliefs or feelings
the hypocrites who criticize other people for not voting but who don't always vote themselves
- M-W
Conformist:
A person who conforms to accepted behavior or established practices.
Source: Oxford Living Dictionaries (English).
My favorite adjective to denote such attitudes is blinkered.
ODO:
blinkered ADJECTIVE
1.1 Having or showing a narrow or limited outlook:
‘a blinkered attitude’‘I'm sorry that you feel I'm so clearly blinkered and entrenched and incapable of having a reasoned discussion about it.’
Another word which works is prejudiced.
ODO:
prejudiced ADJECTIVE
Having or showing a dislike or distrust that is derived from prejudice; bigoted: ‘prejudiced views’
‘Of course, these beliefs do not reflect reality; they reflect the prejudiced views of those who hold them.
prejudice NOUN
1 Preconceived opinion that is not based on reason or actual experience
‘Some prejudices (preconceived opinions of an individual based on opinions about the many) have names such as racism, sexism, or ageism.’
Also, intransigent.
ODO:
intransigent ADJECTIVE
Unwilling or refusing to change one's views or to agree about something:
‘her father had tried persuasion, but she was intransigent’