Let \(G\) be a connected compact semisimple Lie group and \(H,K\) compact subgroups of \(G\) with \(K\subset H\) and \(\dim G>\dim H>\dim K\). This induces a fiber bundle \(M=G/K\to B=G/H\) with typical fiber \(F-H/K\). The Killing form \(\kappa\) of the Lie algebra \({\mathfrak g}\) of \(G\) induces an orthogonal decomposition \({\mathfrak g}={\mathfrak h}\oplus {\mathfrak m}_B= {\mathfrak k}\oplus {\mathfrak m}_F \oplus {\mathfrak m}_B\). The vector space \({\mathfrak m}_F\oplus {\mathfrak m}_B\) can be naturally identified with the tangent space of \(M\) at the origin. Denote by \(\kappa_F\) and \(\kappa_B\) the restriction of \(\kappa\) to \({\mathfrak m}_F\times {\mathfrak m}_F\) and \({\mathfrak m}_B\times {\mathfrak m}_B\), respectively. For each \(a,b>0\) the inner product \(-a\kappa_F- b\kappa_B\) on \({\mathfrak m}_F\oplus{\mathfrak m}_B\) induces a \(G\)-invariant Riemannian metric \(g_{a,b}\) on \(M\). The author classifies all such triples \((G,H,K)\) for which \((G,H)\) is a compact effective irreducible symmetric pair and such that for all \(a,b>0\) every geodesic in \((M,g_{a,b})\) is an orbit of a one-parameter subgroup of \(G\). As an application the author obtains new examples of weakly symmetric spaces. An analogue for non-compact \(G\) is also given.