Income level is important because it will be highly correlated with the amount spent on tutors. If your sample is mainly of higher-income parents (e.g. if you survey parents in-person at a school in an upper-class neighbourhood), than the reported average spend would probably be significantly higher than the average spend in the general population, and so if you multiplied the sampled average by the population size you would have a gross over-estimation of the total market. If you do decide to base your estimate on in-person interviews, make sure your sample includes students/parents from a representative economic status spectrum.
Also note that spend will probably also be highly correlated with student age. I'm guessing parents of high school students will spend more on tutoring than for younger students, because of the higher importance of school performance during high school years.
I agree that paid surveys need to be interpreted critically, as do all surveys and statistical studies. For example, paid survey samples may be skewed towards lower income, leading to an under-estimation of the market size - but for the purpose of validating the market, finding a lower bound for the market size might be good enough (if it's not too low). Also note that paid surveys targeting student will be less skewed than paid surveys targeting parents, because paid surveys are often used to monetise games, and teenagers of higher economic status are similarly likely to prefer filling out a survey over paying with credit card etc.
In regards to the quality of responses, it sounds like you might have had experience with a more qualitative study. For the matter at hand all you really need is a couple of multiple choice numeric-range question, something like: "Thinking of the past 12 months, how much has your family spent on tutoring services in a typical month? * Under $20 * Between $20-$39 * Between $40-$74 * Betweem $75-150 * Over $150" (and a couple of questions about household income level, student age, and any other variables you think may skew your estimate).
For this kind of survey, I doubt the respondents' motivation will have a significant impact on the quality of the results.
I hope that help. Good luck!