We show that in a parametric family of linear recurrence sequences a 1 pαqf 1 pαq n `. . . `ak pαqf k pαq n with the coefficients a i and characteristic roots f i , i " 1, . . . , k, given by rational functions over some number field, for all but a set of α of bounded height in the algebraic closure of Q, the Skolem problem is solvable, and the existence of a zero in such a sequence can be effectively decided. We also discuss several related questions.