Let N=(V,E,c,l) be a network, where G=(V,E) is a directed graph (|V|=n and |E|=m), c(e)<0 is the capacity and l(e)/spl ges/0 is the lead time for each edge e/spl isin/E. The transmission time to send /spl sigma/ units of data from a given source s to a destination t using path p is T(/spl sigma/,p) = l(p) + /spl sigma//c(p), where l(p) is the sum of the lead times of the edges in p, and c(p) is the minimum capacity of the edges in p. The quickest path problem is to find a path of minimum transmission time to transmit the /spl sigma/ units of data from s to t. The problem has applications to fast data transmissions in communication networks. We present parallel algorithms for solving the quickest path problem in the case where the network is sparse [i.e. m=O(n)]. We also give algorithms for solving the dynamic quickest path problem. In this problem, the network, the lead times and the capacities on its edges, as well as the amount of data to be transmitted, change over time. The goal is to build a data structure so that one can very quickly compute the quickest path to transmit a given amount of data from any node s to any node t and also, after a dynamic change (edge lead time or edge capacity modification, or edge deletion) on the input network, to be able to update the data structure in an appropriately short time. Furthermore, we improve upon the best sequential result for the single pair quickest path problem which needs O(rm+rn log n) time, where r is the number of distinct edge capacities.