Measurements are presented of the production of primary K-S(0) and Lambda particles in proton-proton collisions at root s = 7 TeV in the region transverse to the leading charged-particle jet in each event. The average multiplicity and average scalar transverse momentum sum of K-S(0) and Lambda particles measured at pseudorapidities vertical bar eta vertical bar < 2 rise with increasing charged-particle jet p(T) in the range 1-10 GeV/c and saturate in the region 10-50 GeV/c. The rise and saturation of the strange-particle yields and transverse momentum sums in the underlying event are similar to those observed for inclusive charged particles, which confirms the impact-parameter picture of multiple parton interactions. The results are compared to recent tunes of the PYTHIA Monte Carlo event generator. The PYTHIA simulations underestimate the data by 15%-30% for K-S(0) mesons and by about 50% for Lambda baryons, a deficit similar to that observed for the inclusive strange-particle production in non-single-diffractive proton-proton collisions. The constant strange-to charged-particle activity ratios with respect to the leading jet p(T) and similar trends for mesons and baryons indicate that the multiparton-interaction dynamics is decoupled from parton hadronization, which occurs at a later stage.