To increase the speed and sensitivity of the tandem mass spectrometric MS-MS analysis, a tandem of two time-of-flight analyzers (TOFA) is proposed to be used in a novel mode of "nested times": slow separation of parent ions in the first analyzer occurs on the millisecond time scale while fast fragmentation and rapid mass analysis of the fragments in the second analyzer --- on the microsecond scale. The nested time mode allows a so called "parallel analysis" of fragment spectra for all parent ions within a single separation cycle of TOFA1. "Slow" TOFA of a new type are introduced to implement this method. They combine transverse confinement of an ion beam with high-order time-of-flight energy focusing. Radiofrequency (RF) or periodic electrostatic fields are suggested to achieve the transverse confinement. The paper presents performance analysis of the key tandem components including a "slow" TOFA, ion source with an ion storage, and a fragment cell with fast ion transmission.