Children from families whose members have reading impairments are found to be poorer performers, take less advantage of instruction, and require more time to reach the reading level of children whose relatives are good readers. As a family's reading history may not be available, a self-report of reading abilities is used to identify children's background. In this paper, we explored the contribution of phonological, literacy, and linguistic abilities and reported parental reading abilities to predict reading achievement at the end of the school year in a Spanish sample. Children who were starting to read were assessed in a variety of oral language, phonological, and literacy tasks at the beginning and end of the school year. Parents filled out a self-report questionnaire about their reading abilities. Their answers were used to assign children to good or poor reader parent groups (GRP vs PRP). A logistic and ROC analysis were used to assess the variables' discriminative capability, considering literacy scores at the end of the year as a measure of reading achievement. GRP children obtained higher scores than PRP children did. Performance on tasks of rapid naming assessment (RAN) letters (78.6%), Word Reading (75.7%), and Deletion (75.6%) were the most accurate predictors of children's reading achievement. IPRA showed slightly lower accuracy (73.8) than did the behavioral measures and as high specificity as RAN letters (96.2%), similarly to the percentages found in previous studies. Although behavioral measures were shown as the best predictors, parents' self-reports could also provide a quick estimation of family risk of difficulties in literacy acquisition.