It is a good exercise to uphold contrarian views from time to time, so let me begin by offering a few reasons in favor of this form of pseudo-randomization. They are, principally, that it is little different than any other form of systematic sampling, such as obtaining samples of environmental media at points of a grid in the field or sampling every other tree in an orchard, and therefore this sampling might enjoy comparable advantages.
The analogy here is perfect: age was "gridded" by year starting at an origin of zero and assignment to the groups alternated along this (one-dimensional) grid. Some advantages of this approach are to guarantee wide, even dispersion of the sample across the field or orchard (or ages, in this case), which helps even out influences related to location (or time). This can be especially useful when theory suggests that location is the predominant factor in variation of response. Moreover, except for really tiny samples, analyzing the data as if they were a simple random sample introduces relatively little error. Furthermore, some randomization is possible: in the field we can randomly choose the origin and orientation of the grid. In the present case, we can at least randomize whether the even years are controls or treatment subjects.
Another advantage of gridded sampling is to detect localized variation. In the field, this would be "pockets" of unusual responses. Statistically, we may think of them as manifestations of spatial correlation. In the present situation, if there is some chance that a relatively narrow age range experiences unusual responses, then the gridded design is an excellent choice, because a purely randomized design can by chance contain large gaps in ages within one of the groups. (But a better design might be to stratify: use parity of age to form two analytical strata and then, independently within each stratum, randomize patients into control and treatment groups.)
Unfortunately, this defense falls apart once we come to terms with how ages are actually reported. US Census data show that (1) self-reported ages tend to be rounded to multiples of five (I have seen this in analyses of rural block group data) and (2) this tendency is associated with indicators of lower education or socioeconomic status. (It is also well known, although difficult to test, that the final digit in many self-reported ages is $9$, that people in certain fields of work, such as acting, tend to reduce their reported ages, and others will exaggerate their ages for various purposes.) Thus, at least to a slight degree in at least some areas of the US (and even more so elsewhere in the world), the parity of one's reported age is likely to be associated with factors important for the experiment. This renders the concern in the question less than hypothetical: it is real. At this point, the previous answers in this thread capably present the additional thoughts I would care to make, so I will stop and invite you to re-read them.