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She wouldn't take no for an answer about applying for a part-time job.
Literal
She [topic-は] part-time [genitive-の] job [to-に] apply [quotative-と] saying-not-listening.
言ってきかない (or 言って聞かない) is a fixed idiomatic expression literally 'says (it) and doesn't listen' — used to describe someone who stubbornly insists on something and won't be talked out of it. The negative ~ない here is critical: 言ってきく would be a contradiction, but 言ってきかない fuses the speaking and the obstinacy into one verb phrase. The verb 応募する ('apply for') takes に for what one is applying to (a job, contest, programme), paralleling other 'apply / submit' verbs. パート is a Japanese-flavoured loanword shorthand for 'part-time work,' especially the kind of casual/non-career employment historically associated with housewives entering the workforce.