How to say ‘run for office’ in Mandarin?
Both 竞选 and 参选 are common with slight difference in usage. For example:


你会不会参选？ (Will you "enter the election"?)
她会竞选连任。 (She will run for reelection.)


It's hard to pinpoint when one will be favoured over the other, but the examples above will seem weird if the other phrase is used. I think it's due to the literal translation of the phrase: 参选 means "enter the election" and 竞选 means "compete in an election". (or maybe it's just convention that dictates which one is favourable.)

If you want to specify what office one is running for, you can express it as "选 + name of office". For example:


她要选总统。 (She is running for president.)
他要选市长。 (He is running for mayor.)